# Integers that are not the Sum of Positive Powers

Brennan Benfield and Oliver Lippard  Brennan Benfield: Department of
Mathematics and Statistics, University of North Carolina at Charlotte, 9201
University City Blvd., Charlotte, NC 28223, USA  Oliver Lippard: Department of
Mathematics and Statistics, University of North Carolina at Charlotte, 9201
University City Blvd., Charlotte, NC 28223, USA

###### Abstract.

Exactly which positive integers cannot be expressed as the sum of $j$ positive
$k$th powers? This paper utilizes theoretical and computational techniques to
answer this question for $k\leq 9$. Results from Waring’s problem are used
throughout to catalogue the sets of such integers. These sets are then
considered in a general setting, and several curious properties are
established.

The authors extend their utmost appreciation and gratitude to Arindam Roy for
his guidance throughout this project.

Mathematics Subject Classifications. 11P05, 11D09, 11D25, 11P81, 11Y50
Keywords. Waring’s problem, quadratic and cubic Diophantine equations,
partitions

##  1\. Introduction

Diophantus asked if every positive integer could be written as the sum of at
most four perfect squares, which was famously proven by Lagrange [47] in 1770.
Waring [89] quickly proposed the generalization: denote by $g(k)$ the least
number $s$ such that every positive integer is the sum of at most $s$ positive
$k$th powers. In this notation, Lagrange proved that $g(2)=4$. Waring
initially conjectured that $g(3)\leq 9$ and $g(4)\leq 19$, and in general that
$g(k)<\infty$. That $g(k)$ is finite for all $k$ was settled by Hilbert [39]
in 1932. Knowing there are only finitely many positive integers that are not
expressible as the sum of positive $k$th powers begs the question: Exactly
which positive integers cannot be written as the sum of exactly $j$ positive
$k$th powers?

This paper establishes the answer to this question for $3\leq k\leq 9$ and for
sufficiently large $j$. The case for $k=2$ has all but been settled. Further,
this paper examines various properties of the sets of positive integers that
are not expressible as the sum of exactly $j$ positive $k$th powers. The paper
proceeds as follows:

  * •

Section 2 provides preliminary (but necessary for later proofs) information on
Waring’s Problem, on $g(k)$ and $G(k)$, and summarizes the known results for
$k=2$.

  * •

Section 3 establishes new results concerning exactly which positive integers
cannot be expressed as the sum of precisely $j$ positive $k$th powers for
$3\leq k\leq 9$.

  * •

Section 4 extrapolates and generalizes the patterns found in Section 3,
establishing new (and rather surprising) relationships among sets of integers
that have no expression as the sum of $j$ positive $k$th powers.

  * •

The remaining Sections prove the results in Sections 3 and 4 through various
combinatorial and computational methods.

##  2\. Background Information

Small values of $g(k)$ are known; notably Euler’s son, J. A. Euler, [28] is
credited with the current lower bound:
$2^{k}-\lfloor\left(\frac{3}{2}\right)^{k}\rfloor-2\leq g(k)$ which has been
verified for $6\leq k\leq 471600000$ [46, 71]. The conjecture is that this
lower bound is actually the explicit formula for $g(k)$ and has all but been
proven (see Kubina & Wunderlich [46] Mahler [50], Niven [53], Pillai [56],
etc.) The sequence given by $g(k)$ as $k$ increases, begins [61]:

$g(k)=1,4,9,19,37,73,143,279,\ldots\ \qquad\text{for}\ k=1,2,3,\ldots$

Denote by $G(k)$ the least number $s$ such that every sufficiently large
positive integer is the sum of at most $s$ positive $k$th powers. The current
understanding of $G(k)$ is much less complete than of $g(k)$. The only two
known values are $G(2)=4$ (Lagrange in 1770 [47]) and $G(4)=16$ (Davenport in
1939 [16]), though upper and lower bounds exist for several small values:

$4\leq G(3)\leq 7,\quad 6\leq G(5)\leq 17,\quad 9\leq G(6)\leq 24,\quad 8\leq
G(7)\leq 33,\quad 32\leq G(8)\leq 42,$

et cetera. In general, explicit upper bounds for $G(k)$ have been established,
first in 1919 by Hardy and Littlewood [38] who showed that $G(k)\leq(k-2)\cdot
2^{k-1}+5$ using their newly developed circle method. Subsequently, the work
of Vinogradov [79, 80, 81, 82, 83, 84, 85, 86, 87, 88], Tong [74], Chen [13],
Karatsuba [42], Vaughan [75], and Wooley [93], to name a few, led to the
recent publication in 2022 by Brüdern & Wooley [11] that

$G(k)\leq\lceil k(\log k+4.20032)\rceil.$

Lagrange [47] managed to show that $G(2)=4$ despite the fact that there are
infinitely many positive integers that cannot be expressed as the sum of four
positive squares:

$\text{OEIS A000534
\cite[cite]{[\@@bibref{}{OEIS_A000534}{}{}]}}:1,2,3,5,6,8,9,11,14,17,24,29,32,41,56,96,128,224,384,512,896,\ldots$

Lagrange accomplished his proof only because every member of this list is a
sum of $1$ or $2$ or $3$ positive squares. The complete list of such positive
integers was conjectured in 1638 by Descartes [18] and proven in 1911 by
Dubouis [31]. Grosswald [35], elegantly summarizes the result: define the set
$\mathbf{B}^{2}=\\{1,2,4,5,7,10,13\\}$ and denote by $\mathbf{B}_{j}^{k}$ the
set of positive integers that are not the sum of exactly $j$ positive $k$th
powers.

######  Theorem 2.1 (Dubouis).

Every positive integer $n$ is the sum of $4$ nonzero squares provided $n>3$
and $n\neq 4+b$ for $\beta\in\mathbf{B}^{2}\cup\\{25,37\\}$ and $n\neq
2*4^{\alpha},6*4^{\alpha},14*4^{\alpha}$ for $\alpha=0,1,2,\ldots$.

Note that there are infinitely many integers that are not the sum of $4$
positive squares. This is likewise true for sums of $3$ and $2$ positive
square (and obviously $1$ positive square). Grosswald [35] determines the set
of positive integers that are not the sum of $2$ positive squares. Let

$n_{1}=\prod_{p_{i}\equiv 1\mod{4}}p_{i}\qquad\qquad n_{2}=\prod_{p_{i}\equiv
3\mod{4}}p_{i}$

for primes $p_{i}$ where $i=1,2,\ldots$ It is well known [12] that a positive
integer $n$ is the sum of two positive squares if and only if
$n=2^{\alpha}n_{1}n_{2}^{2}$ where $\alpha\geq 0$.

######  Theorem 2.2 (Grosswald [35]).

Let $n=2^{\alpha}n_{1}n_{2}^{2}$ where $\alpha\geq 0$. Then $n$ has a
representation as the sum of two positive squares unless $n_{1}=1$

The complete list of which positive integers cannot be written as the sum of
$j$ positive squares would need to include $j=3$. Unfortunately, this case is
still open and depends on determining all discriminants of binary, positive
definite quadratic forms with exactly one class in each genus (see Chowla
[15]), but there are several known results - notably the work of Hurwitz [40]
and Pall [55]. Synthesizing the contemporary knowledge, Grosswald, Calloway, &
Calloway [36] establish which integers can be expressed as the sum of $3$
positive squares. Define the set $\mathbf{M}$ as the set of positive integers
of the form $n=4^{\alpha}(8m+7)$.

######  Theorem 2.3 (Grosswald, Calloway, & Calloway[36]).

There exists a finite set $\mathbf{T}$ such that every integer
$n\notin\mathbf{M}$ and not of the form $n=4^{\alpha}t$ for $t\in\mathbf{T}$
can be expressed as the sum of three positive squares.

Where the conjectured set $\mathbf{T}=\\{1,2,5,10,13,25,37,58,85,130\\}$ [36].
Weinberger [91] showed that the set $\mathbf{T}$ may contain at most one more
element, which must be greater than $5\cdot 10^{10}$.

For $j\geq 5$, the situation is much different - there are only finitely many
integers that cannot be written as the sum of $j$ positive squares. This seems
like a consequence of Hilbert’s [39] result, that $g(k)<\infty$ (and note that
$G(k)\leq g(k)$), but recall there are infinitely many positive integers that
are not the sum of $4$ positive squares, yet $G(2)=4$. Determining precisely
which integers are not the sum of $j\geq 5$ positive squares was accomplished
by Dubouis [31], Niven [52, 54] and Pall [55]

######  Theorem 2.4 (Dubouis, Niven, Pall).

. Every positive integer $n$ is the sum of $j\geq 6$ nonzero perfect squares
provided $n>j-1$ and $n\neq j+\beta$ for $\beta\in\mathbf{B}^{2}$. For $j=5$,
the result holds for $\mathbf{B}^{2}\cup\\{28\\}$.

This completes the picture for perfect squares, but what happens for higher
powers? The understanding of precisely which positive integers cannot be
expressed as the sum of $j$ positive $k$th powers is not nearly so complete
for $k>2$. This paper makes use of results from Waring’s problem and
Grosswald’s technique for squares in an effort to determine precisely which
positive integers are not the sum of $j$ positive $k$th powers.

Denote by $\mathbf{B}_{j}^{k}$ the set of positive integers that cannot be
expressed as the sum of $j$ positive $k$th powers. For example,

$\mathbf{B}_{5}^{2}=\\{1,2,3,4\\}\cup\\{6,7,9,10,12,15,18,33\\},\quad\mathbf{B}_{6}^{2}=\\{1,2,3,4,5\\}\cup\\{7,8,10,11,13,16,19\\}$

Denote by $\overline{\mathbf{B}_{j}^{k}}$ the set of positive integers that
can be appended to $\mathbf{B}^{k}$ such that every positive integer $n$ is
the sum of $j$ positive $k$th powers provided $n>j-1$ and $n\neq j+\beta$ for
$\beta\in\mathbf{B}^{k}\cup\overline{\mathbf{B}_{j}^{k}}$. For example,
$\overline{\mathbf{B}_{5}^{2}}=\\{28\\}$, and
$\overline{\mathbf{B}_{6}^{2}}=\emptyset$. In other words,

$\mathbf{B}_{j}^{k}=\\{1,2,\ldots,j-1\\}\cup\\{j+\beta\mid\beta\in\mathbf{B}^{k}\cup\overline{\mathbf{B}_{j}^{k}}\\}.$

######  Conjecture 2.5.

For every $k\geq 2$, there exists an index $J^{\prime}$ such that for all
$j\geq J^{\prime}$, all but finitely many positive integers are the sum of
exactly $j$ positive $k$th powers.

Again, Conjecture 2.5 seems obvious from Hilbert’s [39] result that $G(l)\leq
g(k)<\infty$, but recall that $G(2)=4$ even though there are infinitely many
positive integers that are not the sum of $4$ positive squares. It is evident
that $G(k)\leq J^{\prime}$ but equality does not necessarily follow for all
$k$.

######  Conjecture 2.6.

For every $k\geq 2$, there exists an index $J^{*}$ and a finite set
$\mathbf{B}^{k}$ such that every positive integer $n$ is the sum of $j\geq
J^{*}$ positive $k$th powers, provided $n>j-1$ and $n\neq j+\beta$ for
$\beta\in\mathbf{B}^{k}$.

######  Conjecture 2.7.

For $J^{\prime}\leq j<J^{*}$, there exists a finite, nonempty set
$\overline{\mathbf{B}_{j}^{k}}$ such that every positive integer $n$ is the
sum of $j$ positive $k$th powers provided $n>j-1$ and $n\neq j+\beta$ for
$\beta\in\mathbf{B}^{k}\cup\overline{\mathbf{B}_{j}^{k}}$.

##  3\. Results for $3\leq k\leq 9$

###  3.1. Positive Cubes

What positive integers cannot be written as the sum of exactly $j$ positive
cubes? Waring’s original conjecture, that $g(3)=9$ [89], was proven true by
Weiferich [92] and Kempner [44] in the early twentieth century. Work by Landau
[48], Baer [2], and Dickson [29, 26] proved a conjecture by Jacobi in 1851
[41] that $G(3)\leq 8$ where every positive integer other than $23$ and $239$
can be expressed as the sum of at most $8$ positive cubes. Jacobi further
conjectured that $G(3)\leq 7$, and calculated the set $\mathbf{B}_{7}^{3}$.
Subsequent progress was made by Linnik [49], Watson [90], McCurley [51],
Ramaré [57], Bertault [8], Boklan & Elkies [9], and eventually Siksek [59],
who finally proved Jacobi’s conjecture.

######  Theorem 3.1 (Jacobi - Siksek).

Every positive integer other than

$15,22,23,50,114,167,175,186,212,231,238,239,303,364,420,428,454$

can be expressed as the sum of at most $7$ positive cubes.

Define the set

$\displaystyle\mathbf{B}^{3}=\\{$ $\displaystyle
1,2,3,4,5,6,8,9,10,11,12,13,15,16,17,18,19,20,22,23,24,25,27,$ $\displaystyle
29,30,31,32,34,36,37,38,39,41,43,44,45,46,48,50,51,53,55,57,$ $\displaystyle
58,60,62,64,65,67,69,71,72,74,76,79,81,83,86,88,90,93,95,97,$ $\displaystyle
100,102,107,109,114,116,121,123,128,135,142,149\\}.$

######  Theorem 3.2.

Every positive integer $n$ is the sum of $j\geq 14$ positive cubes provided
$n>j-1$ and $n\neq j+\beta$ for $\beta\in\mathbf{B}^{3}$.

######  Theorem 3.3.

For $j=13,12,11,10$ and $9$, every positive integer $n$ is the sum of $j$
positive cubes provided $n>j-1$ and $n\neq j+\beta$ for
$\beta\in\mathbf{B}_{j}^{3}\cup\overline{\mathbf{B}_{j}^{3}}$ where:

$\displaystyle\overline{\mathbf{B}_{13}^{3}}=\\{$ $\displaystyle 212\\}$
$\displaystyle\overline{\mathbf{B}_{12}^{3}}=\\{$ $\displaystyle
186,205,212\\}$ $\displaystyle\overline{\mathbf{B}_{11}^{3}}=\\{$
$\displaystyle 160,179,186,198,205,212,310\\}$
$\displaystyle\overline{\mathbf{B}_{10}^{3}}=\\{$ $\displaystyle
153,160,172,179,186,191,198,205,212,247,284,303,310,364\\}$
$\displaystyle\overline{\mathbf{B}_{9}^{3}}=\\{$ $\displaystyle
153,160,165,172,179,184,186,191,198,205,212,221,238,240,$ $\displaystyle
247,258,277,284,296,301,303,310,338,357,364,413,462\\}$

In fact, for $j=11,10$ and $9$, the sets $\mathbf{B}_{j}^{3}$ are conjectured
in the Online Encyclopedia of Integer Sequences (OEIS) [70], sequences A332111
[69], A332110 [68], and A332109 [67], respectively. For $j=8,7$, and $6$,
there are further conjectures in the OEIS regarding which positive integers
are not the sum of $j$ positive cubes. The following conjecture rewords these
entries in the OEIS in terms of the notation established previously.

######  Conjecture 3.4 (OEIS A332107 [66], A332108 [65], A057907 [64]).

For $j=8,7$ and $6$, every positive integer $n$ is the sum of $j\geq 14$
positive cubes provided $n>j-1$ and $n\neq j+\beta$ for
$\beta\in\mathbf{B}_{j}^{3}\cup\overline{\mathbf{B}_{j}^{3}}$ where:

$\overline{\mathbf{B}_{8}^{3}}=\\{$82, 101, 108, 119, 120, 127, 134, 139, 146,
153, 158, 160, 165, 172, 175, 177, 179, 184, 186, 191, 195, 198, 205, 212,
214, 221, 231, 232, 233, 238, 240, 247, 251, 258, 270, 275, 277, 284, 289,
294, 296, 301, 303, 308, 310, 312, 331, 338, 350, 357, 364, 373, 387, 399,
406, 413, 436, 455, 462, 612$\\}$

$\overline{\mathbf{B}_{7}^{3}}=\\{$56, 75, 82, 94, 101, 108, 112, 113, 119,
120, 127, 132, 134, 139, 146, 151, 153, 158, 160, 165, 168, 169, 170, 172,
175, 177, 179, 184, 186, 188, 191, 195, 198, 205, 206, 207, 212, 214, 221,
224, 225, 226, 231, 232, 233, 238, 240, 244, 245, 247, 249, 251, 258, 263,
268, 270, 275, 277, 282, 284, 286, 287, 289, 294, 296, 301, 303, 305, 308,
310, 312, 323, 324, 331, 336, 338, 343, 347, 350, 357, 361, 364, 366, 373,
380, 387, 392, 397, 399, 406, 410, 413, 421, 429, 432, 436, 447, 448, 455,
462, 485, 488, 495, 504, 523, 549, 579, 586, 605, 612, 636, 914, 940, 947,
970, 996, 1015, 1071, 1087, 1357, 1402, 1572, 1575, 1789, 2023, 2375,
2401$\\}$

$\overline{\mathbf{B}_{6}^{3}}=\\{$49, 56, 68, 75, 82, 87, 94, 101, 105, 106,
108, 112, 113, 119, 120, 125, 127, 132, 134, 139, 143, 144, 146, 151, 153,
158, 160, 161, 162, 163, 165, 168, 169, 170, 172, 175, 177, 179, 180, 181,
182, 184, 186, 188, 191, 195, 198, 199, 200, 205, 206, 207, 212, 214, 217,
218, 219, 221, 223, 224, 225, 226, 231, 232, 233, 237, 238, 240, 242, 244,
245, 247, 249, 251, 256, 258, 260, 261, 263, 268, 270, 273, 275, 277, 279,
280, 282, 284, 286, 287, 289, 294, 296, 297, 298, 301, 303, 305, 308, 310,
312, 316, 317, 321, 323, 324, 329, 331, 334, 335, 336, 338, 340, 343, 347,
350, 354, 357, 358, 359, 361, 364, 366, 369, 371, 373, 380, 384, 385, 387,
390, 392, 395, 397, 399, 403, 406, 410, 413, 414, 421, 422, 425, 429, 432,
436, 440, 441, 447, 448, 455, 459, 460, 462, 469, 476, 478, 481, 485, 486,
488, 495, 497, 502, 504, 512, 516, 521, 523, 542, 547, 549, 553, 560, 567,
572, 573, 576, 579, 586, 593, 598, 605, 610, 612, 628, 629, 636, 638, 647,
648, 654, 664, 665, 673, 674, 693, 699, 703, 725, 727, 729, 732, 745, 755,
764, 765, 781, 790, 800, 816, 823, 827, 844, 846, 847, 851, 856, 863, 872,
877, 882, 884, 888, 889, 891, 907, 908, 914, 921, 933, 940, 944, 945, 947,
952, 954, 963, 970, 973, 982, 989, 996, 1008, 1015, 1024, 1038, 1045, 1060,
1061, 1064, 1071, 1080, 1087, 1097, 1116, 1142, 1143, 1168, 1169, 1187, 1197,
1204, 1230, 1233, 1269, 1278, 1294, 1295, 1331, 1339, 1350, 1357, 1360, 1376,
1386, 1395, 1402, 1405, 1446, 1447, 1448, 1451, 1458, 1483, 1509, 1512, 1529,
1546, 1549, 1565, 1568, 1572, 1574, 1575, 1584, 1628, 1635, 1647, 1665, 1673,
1681, 1691, 1701, 1726, 1744, 1763, 1782, 1789, 1792, 1799, 1808, 1825, 1864,
1871, 1890, 1899, 1906, 1909, 1932, 1951, 1959, 1960, 1962, 1971, 1997, 2016,
2023, 2033, 2059, 2060, 2078, 2104, 2112, 2160, 2167, 2177, 2179, 2186, 2244,
2251, 2267, 2277, 2293, 2296, 2312, 2338, 2349, 2368, 2375, 2384, 2391, 2394,
2401, 2420, 2447, 2475, 2501, 2520, 2560, 2573, 2592, 2654, 2672, 2680, 2762,
2781, 2898, 2961, 3014, 3022, 3059, 3077, 3166, 3168, 3364, 3383, 3401, 3446,
3472, 3653, 3679, 3958, 4092, 4129, 4255, 4300, 4363, 4372, 4382, 4697, 4769,
4876, 4913, 4976, 5182, 5199, 5273, 5299, 5300, 5325, 5497, 5758, 5812, 6181,
6272, 6433, 6694, 7154, 7685, 8036, 19196$\\}$

Also found in the OEIS, the sequence A057906 [63] lists the set
$\mathbf{B}_{5}^{3}$ that was originally conjectured by Romani [58] and the
sequence A057905 [62] lists the set $\mathbf{B}_{4}^{3}$ that was originally
conjectured by Deshouillers, Hennecart, & Landreau [24]. These sets are
becoming too large to display, but will be included for completeness in the
Appendix.

######  Conjecture 3.5 (Romani [58]).

The set $\overline{\mathbf{B}_{5}^{3}}$ has exactly $5360$ elements, the
largest being $1290735$.

######  Conjecture 3.6 (Deshouillers, Hennecart, & Landreau [24]).

The set $\overline{\mathbf{B}_{4}^{3}}$ has exactly $113936597$ elements, the
largest being $7373170279845$.

Note the chain of inclusion among the sets $\overline{\mathbf{B}_{j}^{3}}$.
This is no coincidence, and a generalization of this chain of inclusion is
proven in Section 4.2.

$\ldots\subset\overline{\mathbf{B}_{13}^{3}}\subset\overline{\mathbf{B}_{12}^{3}}\subset\overline{\mathbf{B}_{11}^{3}}\subset\ldots\subset\overline{\mathbf{B}_{2}^{3}}\subset\overline{\mathbf{B}_{1}^{3}}.$

The sets $\mathbf{B}_{3}^{3}$ and $\mathbf{B}_{2}^{3}$ (and of course
$\mathbf{B}_{1}^{3}$) of positive integers that cannot be expressed as the sum
of three or two (or one) positive cubes are known to be infinite. Davenport
[17] showed that every positive integer congruent to $4$ or $5$ modulo $9$
cannot be expressed as the sum of three positive cubes. It is conjectured that
this list is exhaustive. Euler [32] proved the particular case of Fermat’s
last theorem, that no cube is the sum of two positive cubes and Farhi [33]
showed that no (even) perfect number larger than $28$ is the sum two cubes.
Hence, there are infinitely many positive integers that cannot be expressed as
the sum of $1,2$, and $3$ positive cubes. In terms of cardinality,
$\mid\overline{\mathbf{B}_{1}^{3}}\mid=\mid\overline{\mathbf{B}_{2}^{3}}\mid=\mid\overline{\mathbf{B}_{3}^{3}}\mid=\aleph_{0}$.
For a fixed $k$, the size of $\mathbf{B}_{j}^{k}$ creates a sequence for
$j=1,2,\ldots$.

$\mid\mathbf{B}_{j}^{3}\mid=\aleph_{0},\aleph_{0},\aleph_{0},113936676,5439,492,208,142,110,98,92,89,88,89,90,91,92,\ldots$

###  3.2. Positive Fourth Powers

What positive integers are not the sum of exactly $j$ positive fourth powers?
Waring [89] originally conjectured that $g(4)=19$. The first progress was made
by Liouville [37] who proved that $g(4)\leq 53$. Dickson [27] showed that
$g(4)\leq 35$ and Dress [30] showed $g(4)\leq 34$. This was reduced further by
Thomas [73, 72] to $g(4)\leq 22$, but Hardy and Littlewood [38] are credited
with proving that $g(4)\leq 19$. Eventually, spanning a series of papers,
Balasubramanian, Deshouillers, & Dress [3, 4, 22, 19, 20, 21] proved Waring
correct, that $g(4)=19$.

Using their newly developed circle method, Hardy and Littlewood [38] showed
that $G(4)\leq 19$ and shortly afterwards Davenport [16] proved that
$G(4)=16$. Numerical studies by Deshouillers, Hennecart, Kawada, Landreau, and
Wooley [43, 23, 25] collectively determined the set $\mathbf{B}_{16}^{4}$.

######  Lemma 3.7 (Deshouillers, Hennecart, Kawada, Landreau, Wooley).

Every positive integer greater than $13792$ can be expressed as a sum of at
most 16 positive fourths.

Define the set

$\mathbf{B}^{4}=\\{1,2,3,4,5,6,\ldots 2561,2566,2581,2596,2611,2626,2641\\}.$

For powers greater than $3$, the relevant sets begin to grow rapidly to the
point that it is cumbersome to list each set in the body of this paper; data
for powers greater than $3$ are available in the corresponding Appendix. The
full set $\mathbf{B}^{4}$ has $1321$ elements and is listed in the Appendix.

######  Theorem 3.8.

Every positive integer $n$ is the sum of $j\geq 31$ positive fourth powers
provided $n>j-1$ and $n\neq j+b$ for $b\in\mathbf{B}^{4}$.

######  Conjecture 3.9.

Every positive integer $n$ is the sum of $j\geq 21$ positive fourth powers
provided $n>j-1$ and $n\neq j+b$ for $b\in\mathbf{B}^{4}$.

######  Conjecture 3.10.

$\overline{\mathbf{B}_{20}^{4}}=\\{2881,3121,3361,3601,3841,3856,4096,4336,4576,4816\\}$

The set $\overline{\mathbf{B}_{19}^{4}}$ was determined in 1992 by
Deshouillers & Dress [21], and in 2000, Deshouillers, Hennecart, & Landreau
[23] determined the sets $\overline{\mathbf{B}_{18}^{4}},\
\overline{\mathbf{B}_{17}^{4}}$, and $\overline{\mathbf{B}_{16}^{4}}$.

######  Theorem 3.11 (Deshouillers, Dress, Hennecart, & Landreau).

For each $j$ where $16\leq j\leq 20$, there exists a finite, nonempty set
$\overline{\mathbf{B}_{j}^{k}}$ such that every positive integer $n$ is the
sum of $j$ positive fourth powers provided $n>j-1$ and $n\neq j+\beta$ for
$\beta\in\mathbf{B}^{4}$ and $n\notin\overline{\mathbf{B}_{j}^{4}}$.

The sets
$\overline{\mathbf{B}_{16}^{4}},\ldots,\overline{\mathbf{B}_{20}^{4}}$ are
listed in the Appendix. As in the case of cubes, there is a chain of inclusion
among the sets $\overline{\mathbf{B}_{j}^{4}}$.

$\ldots\subset\overline{\mathbf{B}_{20}^{4}}\subset\overline{\mathbf{B}_{19}^{4}}\subset\overline{\mathbf{B}_{18}^{4}}\subset\ldots\subset\overline{\mathbf{B}_{17}^{4}}\subset\overline{\mathbf{B}_{16}^{4}}\subset\overline{\mathbf{B}_{15}^{4}}\subset\ldots\overline{\mathbf{B}_{2}^{4}}\subset\overline{\mathbf{B}_{1}^{4}}.$

For $j\leq 15$, it is known [43] that the sets $\mathbf{B}_{j}^{k}$ are
infinite.

###  3.3. Positive Fifth Powers

What positive integers cannot be expressed as the sum of exactly $j$ positive
fifth powers? Maillet [5] made the first progress in 1896 proving that
$g(5)\leq 192$, followed by Wieferich [92] 1n 1909 who showed that $g(5)\leq
59$, and finally Chen [14] in 1969 who determined that $g(5)=37$.

Vaughan [75] established the upper bound $G(5)\leq 19$ which was then refined
by Brüdern [10] to $G(5)\leq 18$. Vaughan & Wooley [78] eventually reduced the
upper bound to its current status, $G(5)\leq 17$.

######  Lemma 3.12 (Vaughan & Wooley).

Every sufficiently large positive integer can be expressed as a sum of at most
17 positive fifths.

Notice that in the case of fifths (and for all larger cases), the results of
Waring’s problem are established for sufficiently large $n$. Numerical
experimentation indicates the smallest such number.

######  Conjecture 3.13.

Every positive integer greater than $87918$ is the sum of at most $17$
positive fifth powers.

Define the set $\mathbf{B}^{5}$ by

$\mathbf{B}^{5}=\\{1,2,3,4,5,6,7,\ldots,,6137,6168,6199,6230,6261\\}.$

The full set $\mathbf{B}^{5}$ has $3175$ elements and is listed in the
Appendix. Because Vaughan & Wooley’s result [78] is established for
sufficiently large $n$ rather than for all $n>87918$, the proofs for fifths
(and subsequently for all larger powers) is slightly different.

######  Theorem 3.14.

Every (sufficiently large) positive integer is the sum of $j\geq 57$ positive
fifth powers provided $n>j-1$ and $n\neq j+b$ for $b\in\mathbf{B}^{5}$.

######  Theorem 3.15.

Every (sufficiently large) positive integer is the sum of $j$ positive fifth
powers for $23\leq j\leq 56$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{5}\cup\overline{\mathbf{B}_{j}^{5}}$ where
$\overline{\mathbf{B}_{j}^{5}}$ is a finite set given in the Appendix.

######  Conjecture 3.16.

Every positive integer is the sum of $j$ positive fifth powers for $6\leq
j\leq 22$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{5}\cup\overline{\mathbf{B}_{j}^{5}}$ where
$\overline{\mathbf{B}_{j}^{5}}$ is a finite.

As in the case of cubes and fourths, there is a chain of inclusion among the
sets $\overline{\mathbf{B}_{j}^{5}}$ for $6\leq j\leq 57$

$\ldots\subset\overline{\mathbf{B}_{57}^{5}}\subset\overline{\mathbf{B}_{56}^{5}}\subset\overline{\mathbf{B}_{55}^{5}}\subset\ldots\subset\overline{\mathbf{B}_{2}^{5}}\subset\overline{\mathbf{B}_{1}^{5}}.$

###  3.4. Positive Sixth Powers

What positive integers cannot be expressed as the sum of exactly $j$ positive
sixth powers? In 1940, Pillai [56] confirmed J. A. Euler’s conjecture [28])
for sixth powers: proving that
$g(6)=2^{6}+\lfloor\left(\frac{3}{2}\right)^{6}\rfloor-2=73$. Vaughan & Wooley
[78] gave the upper bound for $G(6)$.

######  Lemma 3.17 (Vaughan & Wooley).

Every sufficiently large positive integer can be expressed as a sum of at most
24 positive sixth powers.

Again in the case of sixth powers (and for all larger cases), the results of
Waring’s problem are established for sufficiently large $n$. Numerical
experimentation indicates the smallest such number.

######  Conjecture 3.18.

Every positive integer greater than $1414564$ is the sum of at most $24$
positive sixth powers.

Define the set $\mathbf{B}^{6}$ by

$\mathbf{B}^{6}=\\{1,2,3,4,5,6,7,\ldots,711586,711649\\}.$

The full set $\mathbf{B}^{6}$ has $355825$ elements and is listed in the
Appendix.

######  Theorem 3.19.

Every positive integer $n$ is the sum of $j\geq 86$ positive sixth powers
provided $n>j-1$ and $n\neq j+b$ for $b\in\mathbf{B}^{6}$.

######  Conjecture 3.20.

Every positive integer $n$ is the sum of $j\geq 78$ positive sixth powers
provided $n>j-1$ and $n\neq j+b$ for $b\in\mathbf{B}^{6}$.

######  Conjecture 3.21.

Every positive integer $n$ is the sum of $j$ positive sixth powers for $18\leq
j\leq 77$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{6}\cup\overline{\mathbf{B}_{j}^{6}}$ where
$\overline{\mathbf{B}_{j}^{6}}$ is a finite set, conjectured in the Appendix.

As in the case of cubes, fourths, and fifths, there is a chain of inclusion
among the sets $\overline{\mathbf{B}_{j}^{6}}$:

$\ldots\subset\overline{\mathbf{B}_{77}^{6}}\subset\overline{\mathbf{B}_{76}^{6}}\subset\overline{\mathbf{B}_{75}^{6}}\subset\ldots\subset\overline{\mathbf{B}_{2}^{6}}\subset\overline{\mathbf{B}_{1}^{6}}.$

###  3.5. Positive Seventh Powers

What positive integers cannot be expressed as the sum of exactly $j$ positive
seventh powers? By J. A. Euler’s formula [28]) for seventh powers,
$g(7)=2^{7}+\lfloor\left(\frac{3}{2}\right)^{7}\rfloor-2=143$. Vaughan &
Wooley [78] gave the current upper bound for $G(7)$.

######  Lemma 3.22 (Vaughan & Wooley).

Every sufficiently large positive integer can be expressed as a sum of at most
$33$ positive seventh powers.

Again in the case of seventh powers, the results of Waring’s problem are
established for sufficiently large $n$. Numerical experimentation indicates
the smallest such number.

######  Conjecture 3.23.

Every positive integer greater than $45987024$ is the sum of $33$ positive
seventh powers.

Define the set $\mathbf{B}^{7}$ by

$\mathbf{B}^{7}=\\{1,2,3,4,5,6,7,\ldots,248950,249077\\}.$

The full set $\mathbf{B}^{7}$ has $127839$ elements and is listed in the
Appendix.

######  Theorem 3.24.

Every positive integer $n$ is the sum of $j\geq 246$ positive seventh powers
provided $n>j-1$ and $n\neq j+b$ for $b\in\mathbf{B}^{7}$.

######  Theorem 3.25.

Every positive integer $n$ is the sum of $j$ positive seventh powers for
$245\geq j\geq 51$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{7}\cup\overline{\mathbf{B}_{j}^{7}}$ where
$\overline{\mathbf{B}_{j}^{7}}$ is a finite set, listed in the Appendix.

######  Conjecture 3.26.

Every positive integer $n$ is the sum of $j$ positive seventh powers for
$51\geq j\geq 25$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{7}\cup\overline{\mathbf{B}_{j}^{7}}$ where
$\overline{\mathbf{B}_{j}^{7}}$ is a finite set, conjectured in the Appendix.

As in the previous cases, there is a chain of inclusion among the sets
$\overline{\mathbf{B}_{j}^{7}}$ for $8\leq j\leq 245$

$\ldots\subset\overline{\mathbf{B}_{245}^{7}}\subset\overline{\mathbf{B}_{244}^{7}}\subset\overline{\mathbf{B}_{243}^{7}}\subset\ldots\subset\overline{\mathbf{B}_{2}^{7}}\subset\overline{\mathbf{B}_{1}^{7}}.$

###  3.6. Positive Eighth Powers

What positive integers cannot be expressed as the sum of exactly $j$ positive
eighth powers? By J. A. Euler’s formula [28]) for eighth powers,
$g(8)=2^{8}+\lfloor\left(\frac{3}{2}\right)^{8}\rfloor-2=279$. Define the set
$\mathbf{B}^{8}$ by

$\mathbf{B}^{8}=\\{1,2,3,4,5,6,7,\ldots,1889986,1890241\\}.$

The full set $\mathbf{B}^{8}$ has $945121$ elements and is listed in the
Appendix. Vaughan & Wooley [78] gave the current upper bound for $G(8)$.

######  Lemma 3.27 (Vaughan & Wooley).

Every sufficiently large positive integer can be expressed as a sum of at most
$42$ positive eighth powers.

Again in the case of eighth powers, the results of Waring’s problem are
established for sufficiently large $n$. Unlike the previous cases, computation
to establish the smallest such number is out of reach for modest computers.

######  Question 3.28.

What is the largest positive integer that is not the sum of at most $42$
positive eighth powers?

The nature of the proof technique used in the later sections requires an exact
value. Because the largest positive integer that is not the sum of at most
$42$ positive eighth powers is unattainable, the following (slightly weaker)
result is used instead:

######  Conjecture 3.29.

Every positive integer greater than $904339959$ is the sum of exactly $47$
positive eighth powers.

######  Theorem 3.30.

Every positive integer $n$ is the sum of $j\geq 334$ positive eighth powers
provided $n>j-1$ and $n\neq j+b$ for $b\in\mathbf{B}^{8}$.

######  Theorem 3.31.

Every positive integer $n$ is the sum of $j$ positive eighth powers for
$69\leq j\leq 333$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{8}\cup\overline{\mathbf{B}^{8}_{j}}$ where
$\overline{\mathbf{B}^{8}_{j}}$ is a finite set, conjectured in the Appendix

######  Theorem 3.32.

Every positive integer $n$ is the sum of $j$ positive eighth powers for
$64\leq j\leq 68$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{8}\cup\overline{\mathbf{B}_{j}^{8}}$ where
$\overline{\mathbf{B}_{j}^{8}}$ is a finite set, conjectured in the Appendix.

######  Conjecture 3.33.

Every positive integer is the sum of $j$ positive eighth powers for $32\leq
j\leq 63$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{8}\cup\overline{\mathbf{B}_{j}^{8}}$ where
$\overline{\mathbf{B}_{j}^{8}}$ is a finite set given in the Appendix.

As in the previous cases, there is a chain of inclusion among the sets
$\overline{\mathbf{B}_{j}^{8}}$:

$\ldots\subset\overline{\mathbf{B}_{334}^{8}}\subset\overline{\mathbf{B}_{333}^{8}}\subset\overline{\mathbf{B}_{332}^{8}}\subset\ldots\subset\overline{\mathbf{B}_{2}^{8}}\subset\overline{\mathbf{B}_{1}^{8}}.$

###  3.7. Positive Ninth Powers

What positive integers cannot be expressed as the sum of exactly $j$ positive
ninth powers? By J. A. Euler’s formula [28]),
$g(9)=2^{9}+\lfloor\left(\frac{3}{2}\right)^{9}\rfloor-2=548$. Vaughan &
Wooley [78] gave the current upper bound for $G(9)$.

######  Lemma 3.34 (Vaughan & Wooley).

Every sufficiently large positive integer can be expressed as a sum of at most
$50$ positive ninth powers.

Again in the case of ninth powers, the results of Waring’s problem are
established for sufficiently large $n$. Unlike the previous cases, computation
to establish the smallest such number is out of reach for modest computers.

######  Question 3.35.

What is the largest positive integer that is not the sum of at most $50$
positive ninth powers?

The nature of the proof technique used in the later sections requires an exact
value. Because the largest positive integer that is not the sum of at most
$50$ positive ninth powers is unattainable, the following (slightly weaker)
result is used instead:

######  Conjecture 3.36.

Every positive integer greater than $967214052$ is the sum of exactly $121$
positive ninth powers.

Define the set $\mathbf{B}^{9}$ by

$\mathbf{B}^{9}=\\{1,2,3,4,5,6,7,\ldots,6463886,6464397\\}.$

The full set $\mathbf{B}^{9}$ has $3438463$ elements and is listed in the
Appendix.

######  Theorem 3.37.

Every positive integer $n$ is the sum of $j\geq 717$ positive ninth powers
provided $n>j-1$ and $n\neq j+b$ for $b\in\mathbf{B}^{9}$.

######  Theorem 3.38.

Every positive integer $n$ is the sum of $j$ positive ninth powers for
$188\leq j\leq 716$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{9}\cup\overline{\mathbf{B}^{9}_{j}}$ where
$\overline{\mathbf{B}^{9}_{j}}$ is a finite set, conjectured in the Appendix

######  Conjecture 3.39.

Every positive integer is the sum of $j$ positive ninth powers for $121\leq
j\leq 187$, provided $n>j-1$ and $n\neq j+b$ for
$b\in\mathbf{B}^{9}\cup\overline{\mathbf{B}_{j}^{9}}$ where
$\overline{\mathbf{B}_{j}^{9}}$ is a finite set given in the Appendix.

As in the previous cases, there is a chain of inclusion among the sets
$\overline{\mathbf{B}_{j}^{9}}$:

$\ldots\subset\overline{\mathbf{B}_{717}^{9}}\subset\overline{\mathbf{B}_{716}^{9}}\subset\overline{\mathbf{B}_{715}^{9}}\subset\ldots\subset\overline{\mathbf{B}_{2}^{9}}\subset\overline{\mathbf{B}_{1}^{9}}.$

##  4\. Properties of $\mathbf{B}^{k}$ and $\mathbf{B}_{j}^{k}$

###  4.1. The size of $\mathbf{B}^{k}$

Consider the sequence created by the number of elements in $\mathbf{B}^{k}$
for $k=2,3,\ldots$ Define the sequence $a_{k}$ to be the maximum of
$\mathbf{B}^{k}$ for $k\geq 2$, and define the sequence
$b_{k}=\mid\mathbf{B}^{k}\mid$. It is curious that in many cases, but not all,
the largest element in $\mathbf{B}^{k}$ is twice and one less than the size of
$\mathbf{B}^{k}$.

$k$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---
$a_{k}$ | 13 | 149 | 2641 | 6261 | 711649 | 249077 | 1890241 | 6464397
$b_{k}$ | 7 | 75 | 1321 | 3175 | 355825 | 127839 | 945121 | 3438463
Table 1. $a_{k}$ and $b_{k}$ for $2\leq k\leq 9$

######  Conjecture 4.1.

For all $k,a_{2^{k}}=2b_{2^{k}}-1$.

######  Conjecture 4.2.

$n\in\mathbf{B}^{2^{k}}$ if and only if
$a_{2^{k}}-n\not\in\mathbf{B}^{2^{k}}$.

######  Conjecture 4.3.

$\lim_{k\rightarrow\infty}\frac{a_{k}}{b_{k}}=2$.

######  Conjecture 4.4.

$a_{k}$ and $b_{k}$ are odd for all $k$.

Note that Conjecture 4.2 implies Conjecture 4.1.

######  Conjecture 4.5.

The sequence $\mid\mathbf{B}^{k}\mid$ for $k=2,3,\ldots$ never becomes
monotone.

######  Theorem 4.6.

For any odd prime $p,\mid\mathbf{B}^{p-1}\mid\geq
p^{p-1}\left(\frac{p-1}{2}\right)$.

###### Proof.

The number of integers $n>k$ such that there are no solutions to equation

$\displaystyle\sum_{i=1}^{j}a_{i}^{p-1}=n$

is by definition the size of the set $\mathbf{B}^{p-1}$. Recall Fermat’s
Little Theorem: unless $p|a_{i}$, it must be that $a_{i}^{p-1}\equiv
1\pmod{p}$ for any odd prime $p$. For $n<rp^{p-1}$, at most $r-1$ of the
$a_{i}$’s can be multiples of $p$, so the sum will be in the same equivalence
class modulo $p$ as at least one of the integers in the interval $[k,k+r-1]$.
This leaves $p-r$ equivalence classes (modulo p) that cannot be expressed as a
sum of exactly $j$ positive $p-1$ powers with a sum of less than $rp^{p-1}$.

There are $p^{p-2}$ positive integers between $(r-1)p^{p-1}$ and $rp^{p-1}$ in
each equivalence class modulo $p$. Hence, there are at least $(p-r)p^{p-2}$
integers in the interval $((r-1)p^{p-1},rp^{p-1}]$ that cannot be expressed as
a sum of $j$ positive $p-1^{\textsuperscript{th}}$ powers. Summing over all
$r$ such that $1\leq r\leq p$ gives a lower bound for the size of
$\mathbf{B}^{p-1}$:

$\mid\mathbf{B}^{p-1}\mid\ \geq\sum_{r=1}^{p}(p-r)p^{p-2}.$

Substituting $s=p-r$ gives

$\mid\mathbf{B}^{p-1}\mid\ \geq\sum_{s=0}^{p-1}sp^{p-2}=\frac{p(p-1)}{2}\cdot
p^{p-2}=p^{p-1}\left(\frac{p-1}{2}\right).$

∎

######  Corollary 4.7.

$\mid\mathbf{B}^{10}\mid\geq 129687123005$.

###  4.2. Chain of Inclusion: $\mathbf{B}_{j+1}^{k}\subset\mathbf{B}_{j}^{k}$

As seen in the particular cases for $3\leq j\leq 9$, there is a relationship
among the sets $\overline{\mathbf{B}_{j}^{k}}$.

######  Theorem 4.8.

For all $j,k\in\mathbb{N}$,
$\overline{\mathbf{B}_{j}^{k}}\subseteq\overline{\mathbf{B}_{j}^{k+1}}$

The proof of Theorem 4.8 utilizes the Theory of Partitions. The partition
function $p(n)$ enumerates the number of partitions of a positive integer $n$
where the partitions are positive integer sequences
$\lambda=(\lambda_{1},\lambda_{2},...)$ with
$\lambda_{1}\geq\lambda_{2}\geq~{}\dots>0$ and $\sum_{j\geq
1}{\lambda_{j}}=n$. For example, $p(4)=5$ since the only ways to partition $4$
are $4$, $3+1$, $2+2$, $2+1+1$, and $1+1+1+1$.

It has become increasingly popular to consider restricted partition functions,
typically denoted $p_{A}(n)$ where $A$ is some constraint on $\lambda$. For
instance, the integer $4$ could be partitioned using only prime numbers, only
positive squares, only powers of $2$, only odd numbers, etc:

$\displaystyle 4=2+2$ $\displaystyle\ \ \qquad 4$ $\displaystyle=2^{2}$
$\displaystyle\ \ \qquad 4$ $\displaystyle=2^{2}$ $\displaystyle\ \ \qquad 4$
$\displaystyle=3+1$ $\displaystyle=1^{2}+1^{2}+1^{2}+1^{2}$
$\displaystyle=2^{1}+2^{1}$ $\displaystyle=1+1+1+1$
$\displaystyle=2^{0}+2^{0}+2^{0}+2^{0}$

There are two restricted functions that are of particular interest regarding
generalized taxicab numbers: The $k$th power partition function and the
partitions into exactly $j$ parts function.

The $k$-th power partition function (see [7, 34, 76, 94]), denoted $p^{k}(n)$,
counts the number of ways that a positive integer $n$ can be written as the
sum of positive $k$-th powers. Note that $p^{k}(n)=p(n)$ when $k=1$, and for
$k=2$, $p^{2}(n)$ restricts $\lambda$ to positive squares. For example,
$p^{2}(4)=2$ since $4=2^{2}=1^{2}+1^{2}+1^{2}+1^{2}$ and $p^{k}(4)=1$ for all
$k\geq 3$.

The partitions into exactly $j$ parts function (see [1, 45]), denoted $p(n,j)$
counts the number of ways that a positive integer $n$ can be written as a sum
of exactly $j$ integers. For example, $p(4,2)=2$ because there are exactly two
ways to write $4$ as a sum of $2$ integers: $4=3+1=2+2$.

Synthesizing the combinatorial properties of these two restricted partition
functions creates the $k$th power partition into exactly $j$ parts function,
denoted $p^{k}(n,j)$. In a previous paper, the authors along with Arindam Roy
[6] established a useful identity for this partition function:

######  Lemma 4.9.

For all $n,j\geq 2$,

$p^{k}(n,j)\leq p^{k}(n+1,j+1)$

with equality whenever $n<2^{k}j$.

###### Proof of Theorem 4.8.

The set $\overline{\mathbf{B}_{j}^{k}}$ is the set of positive integers $b$
such that $b-j$ has zero representations as a sum of $j$ positive
$k^{\textsuperscript{th}}\ $ powers. For some positive integer $c$, suppose
$c\not\in\overline{\mathbf{B}_{j}^{k}}$. If $p^{k}(c+j,j)=0$, then
$c\in\overline{\mathbf{B}_{j}^{k}}$, contradicting the hypothesis. Thus,
$p^{k}(c+j,j)\geq 1$ for all $c$. It follows from Lemma 4.9 that
$p^{k}(c+(j+1),j+1)\geq 1$, hence $c\not\in\overline{\mathbf{B}_{j+1}^{k}}$.
Then, if $c\in\overline{\mathbf{B}_{j+1}^{k}}$, then
$c\in\overline{\mathbf{B}_{j}^{k}}$. ∎

######  Theorem 4.10.

If $a_{k}<j(2^{k}-1)$, then
$\overline{\mathbf{B}_{j}^{k}}=\overline{\mathbf{B}_{j+1}^{k}}$.

###### Proof.

By Theorem 4.8,
$\overline{\mathbf{B}_{j+1}^{k}}\subseteq\overline{\mathbf{B}_{j}^{k}}$. It
remains to show that
$\overline{\mathbf{B}_{j}^{k}}\subseteq\overline{\mathbf{B}_{j+1}^{k}}$.
Suppose $n\in\overline{\mathbf{B}_{j}^{k}}$. Then, $p^{k}(n+j,j)=0$. Because
$n<j(2^{k}-1)$, it follows that $n+j<j(2^{k})$. Then,
$p^{k}(n+j,j)=p^{k}(n+(j+1),j+1)=0$. Hence,
$n\in\overline{\mathbf{B}_{j}^{k}}$. ∎

A useful corollary follows: it is only necessary to check up to
$j=\frac{a_{k}}{2^{k}-1}$ to be sure that a candidate set is in fact the
complete set $\mathbf{B}^{k}$.

######  Corollary 4.11.

For all $j\geq\frac{a_{k}}{2^{k}-1}$, the set
$\overline{\mathbf{B}_{j}^{k}}=\emptyset$.

Note that if $\overline{\mathbf{B}_{j}^{k}}=\emptyset$, then
$\overline{\mathbf{B}_{j+i}^{k}}=\emptyset$ for $i=1,2,\ldots,9$ Define the
sequence $\\{x_{k}\\}_{k\in\mathbb{N}}$, where $x_{k}:=\min\\{j:\
\overline{\mathbf{B}_{j}^{k}}=\emptyset\\}$. The sequence $x_{k}$ for
$k=1,2,\ldots$ begins:

$x_{k}=1,6,15,22,58,78,244,334,717,\ldots$

######  Conjecture 4.12.

The sequence $x_{k}$ is strictly increasing.

######  Conjecture 4.13.

The value of $x_{k}$ is odd if and only if $k$ is of the form $3^{\alpha}$ for
$\alpha=0,1,\ldots$

######  Question 4.14.

Is it possible to find an explicit formula or an asymptotic for the sequence
$x_{k}$?

###  4.3. Chain of Equality

######  Lemma 4.15.

If $n$ is a positive integer, then $n\not\in\mathbf{B}^{k}_{j+1}$ if and only
if there is no positive $b$ such that $n-b^{k}\not\in\mathbf{B}^{k}_{j}$.

###### Proof.

Let $n$ be a positive integer; by definition, $n\in\mathbf{B}^{k}_{j}$ if and
only if there is no way to write $n$ as a sum of $j$ positive
$k\textsuperscript{th}\ $ powers. Suppose $n\not\in\mathbf{B}^{k}_{j+1}$.
Then, there must be a representation of $n$ as a sum of $j+1$ positive
$k\textsuperscript{th}\ $ powers, which can be split into one
$k\textsuperscript{th}\ $ power and a sum of $j$ positive
$k\textsuperscript{th}\ $ powers. Hence, there exists some $b$ such that $n-b$
yields some positive integer that is the sum of $j$ positive
$k\textsuperscript{th}\ $ powers, that is, a positive integer that is not in
the set $\mathbf{B}^{k}_{j}$. To prove the converse, suppose
$n\in\mathbf{B}^{k}_{j+1}$. If there is a positive $b$ such that
$n-b^{k}\not\in\mathbf{B}^{k}_{j}$, then adding $b^{k}$ yields a
representation of $n$ as a sum of $j+1$ positive $k\textsuperscript{th}\ $
powers. In other words, $n\in\mathbf{B}^{k}_{j}$, which contradicts the
hypothesis. Hence, there is no positive $b$ such that
$n-b^{k}\not\in\mathbf{B}^{k}_{j}$. ∎

######  Lemma 4.16.

If $\max\left(\mathbf{B}^{k}_{j}\right)<2^{k}(j+1)$, then
$\overline{\mathbf{B}_{k}^{j}}=\emptyset$.

###### Proof.

This is a consequence of Theorem 4.10 under particular initial conditions. ∎

######  Lemma 4.17.

Let $m=\max\left(\mathbf{B}^{k}_{j}\right)$. Suppose $\mathbf{B}^{k}_{j+1}=$
{1} $\cup$ {$n+1:n\in\mathbf{B}^{k}_{j}$} and
$\left\lfloor\sqrt[k]{m+1}\right\rfloor=\left\lfloor\sqrt[k]{m}\right\rfloor$.
Then, $\mathbf{B}^{k}_{j+2}=$ {1} $\cup$ {$n+1:n\in\mathbf{B}^{k}_{j+1}$}.

###### Proof.

By Lemma 4.15, it is only necessary to check values of $b$ where $b^{k}<n$,
since a negative number can never have a representation as a sum of positive
powers. Hence, the maximum value of $b$ for which Lemma 4.15 applies is
$\lfloor\sqrt[k]{m}\rfloor$. Suppose $a\in\mathbf{B}^{k}_{j}$. Then,
$a+1\in\mathbf{B}^{k}_{j+1}$ by assumption, and by Lemma 4.15 there exists
some positive integer $1\leq b\leq\lfloor\sqrt[k]{m}\rfloor$, such that
$a+1-b^{k}\not\in\mathbf{B}^{k}_{j}$. By assumption,
$a+1-b^{k}\not\in\mathbf{B}^{k}_{j+1}$, implying $a+2\in\mathbf{B}^{k}_{j+2}$.
On the contrary, if $a\not\in\mathbf{B}^{k}_{j}$, then there is no positive
$b$ such that $a-b^{k}\not\in\mathbf{B}^{k}_{j}$. Because $b$ is arbitrary,
$1\leq b\leq\lfloor\sqrt[k]{m}\rfloor$. As with the previous case,
$a+1-b^{k}\not\in\mathbf{B}^{k}_{j+1}$ by assumption, so there is again no
positive $b,1\leq b\leq\lfloor\sqrt[k]{m}\rfloor$, such that
$a+1-b^{k}\not\in\mathbf{B}^{k}_{j+1}$. However,
$\max\left(\mathbf{B}^{k}_{j+1}\right)=m+1$, so it is now necessary to check
values of $b$ up to $\lfloor\sqrt[k]{m+1}\rfloor$. Since
$\lfloor\sqrt[k]{m+1}\rfloor=\lfloor\sqrt[k]{m}\rfloor$ by assumption, there
are no other possible values of $b$, and so $a+2\not\in\mathbf{B}^{k}_{j+2}$.
Because $j$ is positive, $j+2\geq 3$, and there is no way to write $1$ or $2$
as a sum of $j$ positive $k\textsuperscript{th}\ $ powers, completing the
proof. ∎

######  Theorem 4.18.

Let $m=\max\left(\mathbf{B}^{k}_{j}\right)$. Suppose $\mathbf{B}^{k}_{j+1}=$
{1} $\cup$ {$n+1:n\in\mathbf{B}^{k}_{j}$} and
$\left\lfloor\sqrt[k]{m+\frac{m-j}{2^{k}-1}-j}\right\rfloor=\left\lfloor\sqrt[k]{m}\right\rfloor$.
Then, $\mathbf{B}^{k}=$ {$n-j:n\in\mathbf{B}^{k}_{j},n>j$}, and
$\overline{\mathbf{B}^{k}_{l}}=\emptyset$ for all $l\geq j$.

###### Proof.

By Lemma 4.17,

$\mathbf{B}^{k}_{j+2}=\\{1\\}\cup\\{n+1:n\in\mathbf{B}^{k}_{j+1}\\}=\\{1,2\\}\cup\\{n+2:n\in\mathbf{B}^{k}_{j}\\}.$

Similarly,

$\mathbf{B}^{k}_{j+3}=\\{1,2,3\\}\cup\\{n+3:n\in\mathbf{B}^{k}_{j}\\},\mathbf{B}^{k}_{j+4}=\\{1,2,3,4\\}\cup\\{n+4:n\in\mathbf{B}^{k}_{j}\\}.$

This process continues inductively until

$\mathbf{B}^{k}_{\left\lceil\frac{m-j}{2^{k}-1}\right\rceil}=\\{1,2,3,\ldots,\left\lceil\frac{m-j}{2^{k}-1}\right\rceil+1-j\\}\cup\\{n+\left\lceil\frac{m-j}{2^{k}-1}\right\rceil-j:n\in\mathbf{B}^{k}_{j}\\}$

of which the largest element is

$m+\left\lceil\frac{m-j}{2^{k}-1}\right\rceil-j<m-j+\frac{m-j}{2^{k}-1}+1=(m-j)\left(\frac{2^{k}}{2^{k}-1}\right)+1<2^{k}\left(\left\lfloor\frac{m-j}{2^{k}-1}\right\rfloor+1\right).$

By Lemma 4.16, for any larger positive integer $l\geq j$, it follows that

$\mathbf{B}^{k}_{l+1}=\\{1\\}\cup\\{n+1:n\in\mathbf{B}^{k}_{l}\\}.$

Since the only difference between $\mathbf{B}^{k}_{l}$ and
$\mathbf{B}^{k}_{l+1}$ is the addition of $1$ to each term of
$\mathbf{B}^{k}_{l}$, it follows that the set $\mathbf{B}^{k}=$
{$n-j:n\in\mathbf{B}^{k}_{j},n>j$} by definition, and that
$\overline{\mathbf{B}^{k}_{l}}=\emptyset$. ∎

For example, Grosswald’s proof of $\mathbf{B}^{2}$ can be restated using the
fact that $\mathbf{B}^{2}_{6}=$ {1, 2, 3, 4, 5} $\cup$ {7, 8, 10, 11, 13, 16,
19} and $\mathbf{B}^{2}_{7}=$ {1, 2, 3, 4, 5, 6} $\cup$ {8, 9, 11, 12, 14, 17,
20}. In this case, $m=19,j=6$, and $k=2$. The second condition in Theorem 4.18
becomes

$\left\lfloor\sqrt{13+\frac{13}{3}}\right\rfloor=\left\lfloor\sqrt{19}\right\rfloor.$

###  4.4. Density of numbers expressible as a sum of $j$ positive
$k\textsuperscript{th}$ powers

How sparsely populated among all positive integers are those that have a
representation as the sum of exactly $j$ positive $k\textsuperscript{th}$
powers? Let

$S_{j}^{k}(n)=\\{i\leq n\ \text{$\mid i$ can be expressed as a sum of exactly
$j$ positive $k\textsuperscript{th}$ powers}\\}$

and define the asymptotic density of such expressible numbers as

$d_{j}^{k}=\lim_{n\rightarrow\infty}\frac{\mid S_{j}^{k}(n)\mid}{n}.$

######  Theorem 4.19.

For all positive integers $j<k$, $\lim_{n\rightarrow\infty}\frac{\mid
S_{j}^{k}(n)\mid}{n}=0$.

###### Proof.

Suppose $m\in\mathbb{N}$ is expressible as a a sum of $j$ positive
$k\textsuperscript{th}$ powers, i.e. $m=\sum_{i=1}^{j}a_{i}^{k}$. Then, the
maximum possible $a_{i}$ is $\sqrt[k]{m}$. Thus, there are
$\left(\sqrt[k]{m}\right)^{j}=m^{j/k}$ possible combinations of positive
$k\textsuperscript{th}$ powers that may lead to a sum less than or equal to
$m$. Note that some of these combinations may have sums greater than $m$ or
lead to the same sum in two different ways. Fortunately, this is still enough
to establish the upper bound $\mid S_{j}^{k}(m)\mid\leq m^{j/k}$:

$d_{j}^{k}=\lim_{n\rightarrow\infty}\frac{\mid
S_{j}^{k}(n)\mid}{n}\leq\lim_{n\rightarrow\infty}\frac{m^{j/k}}{m}\leq\lim_{n\rightarrow\infty}m^{j/k-1}=0$

since by assumption, $j<k$ implying that $(j/k)-1$ is negative. ∎

With a similar notion to the previous definitions, define for a positive
integer $i$,

$\sigma_{j}^{k}(n)=\\{i\leq n:i\ \text{can be expressed as a sum of {at
most}}\ j\ \text{positive}\ k\textsuperscript{th}\ \text{powers}\\}.$

######  Corollary 4.20.

For all $j<k$,
$\lim_{n\rightarrow\infty}\frac{\mid\sigma_{j}^{k}(n)\mid}{n}=0$.

###### Proof.

The asymptotic density of expressible numbers is

$\delta_{j}^{k}=\lim_{n\rightarrow\infty}\frac{\mid\sigma_{j}^{k}(n)\mid}{n}.$

Then, $\sigma_{j}^{k}(n)=\bigcup_{i=0}^{j}S_{j}^{k}(n)$. Hence,
$\delta_{j}^{k}\leq\sum_{i=0}^{j}d_{j}^{k}(n)=0$. ∎

######  Proposition 4.21.

At most $\frac{1}{k}$ of all positive integers are expressible as the sum of
$j$ positive $k\textsuperscript{th}\ $ powers, if $j=k$.

###### Proof.

Again suppose $n\in\mathbb{N}$ is expressible as a sum of $j$ positive
$k\textsuperscript{th}\ $ powers, with $n=\sum_{i=1}^{j}a_{i}^{k}$. Every
power in the representation of $n$ must be less than $n$, which leaves
$\left(\sqrt[k]{n}\right)$ possibilities. There are
$\left(\sqrt[k]{n}\right)^{j}=\left(\sqrt[k]{n}\right)^{k}=n$ combinations of
$j$ positive $k\textsuperscript{th}\ $ powers that could possibly be less than
$n$, with representations of different order (for example, $2^{2}+3^{2}$ and
$3^{2}+2^{2}$) counting as distinct, despite them giving the same sum. Out of
these, $\left(\sqrt[k]{n}\right)$ will have all identical terms, while the
remainder will have at least one term different from one other term. The
former will vanish as $n\rightarrow\infty$, while the latter must have at
least $j=k$ possible permutations. Dividing by the number of permutations
gives the amount of permutations when order does not matter, which is thus at
most $\frac{1}{k}$ of positive integers as $n$ becomes arbitrarily large. ∎

######  Proposition 4.22.

For all positive integers $k$, a positive proportion (at least
$\frac{1}{g(k)}$) of integers can be written as a sum of $n$ positive
$k\textsuperscript{th}$ powers for all $n\geq g(k)$, where $g(k)$ is the
solution to Waring’s problem for $k\textsuperscript{th}$ powers.

###### Proof.

By the result of Waring’s Problem, every positive integer is the sum of at
most $g(k)$ perfect $k\textsuperscript{th}$ powers. Let $n$ be a sum of
exactly $m$ positive $k\textsuperscript{th}$ powers for some $m\leq g(k)$.
Then, $n+g(k)-m$ is a sum of exactly $g(k)$ positive $k\textsuperscript{th}$
powers, obtained by adding $n-m$ copies of $1^{k}$ to the original sum. Since
$0\leq m\leq g(k)$, then there is some $l$ such that $n\leq l<n+g(k)$ such
that $l$ can be represented as a sum of exactly $g(k)$ perfect
$k\textsuperscript{th}$ powers. Since for every $n$, at least one positive
integer between $n$ and $n+g(k)-1$, inclusive, must be expressible as a sum of
exactly $g(k)$ positive $k\textsuperscript{th}$ powers, the proportion of
positive integers that can be expressed as a sum of exactly $g(k)$ positive
$k^{\textsuperscript{th}}$ powers is at least $\frac{1}{g(k)}$. ∎

######  Conjecture 4.23.

For any $\kappa\geq 2$ and $j<2^{\kappa+2}$, there are infinitely many
positive integers that cannot be represented as a sum of $j$ positive $k$th
powers, where $k=2^{\kappa}$.

##  5\. Proof of Theorems 3.2 & 3.3

###### Proof.

Observe that the integer $1072$ has the marvelous property that it is the sum
of $2,3,4,5,6,7$, and $8$ positive cubes, and many more cubes.

$\displaystyle 1072$ $\displaystyle=9^{3}+7^{3}$
$\displaystyle=10^{3}+4^{3}+2^{3}$ $\displaystyle=8^{3}+7^{3}+6^{3}+1^{3}$
$\displaystyle=8^{3}+6^{3}+6^{3}+4^{3}+4^{3}$
$\displaystyle=7^{3}+7^{3}+7^{3}+3^{3}+2^{3}+2^{3}$
$\displaystyle=7^{3}+6^{3}+6^{3}+6^{3}+3^{3}+3^{3}+3^{3}$
$\displaystyle=7^{3}+6^{3}+6^{3}+6^{3}+4^{3}+2^{3}+2^{3}+1^{3}$

Let $n>1072+454=1526$ and consider when $j=9$. By Lemma 3.1,
$n-1072=\sum_{i=1}^{7}x_{i}^{3}$ where $x_{7}\geq\ldots\geq x_{1}\geq 0$. If
all $x_{i}$’s are positive, then $n=9^{3}+7^{3}+\sum_{i=1}^{7}x_{i}^{3}$. If
one of the $x_{i}$’s is zero, then
$n=10^{3}+4^{3}+2^{3}+\sum_{i=1}^{6}x_{i}^{3}$. If two of the $x_{i}$’s are
zero, then $n=8^{3}+7^{3}+6^{3}+1^{3}+\sum_{i=1}^{5}x_{i}^{3}$. If three of
the $x_{i}$’s are zero, then
$n=8^{3}+6^{3}+6^{3}+4^{3}+4^{3}+\sum_{i=1}^{4}x_{i}^{3}$. If four of the
$x_{i}$’s are zero, then
$n=7^{3}+7^{3}+7^{3}+3^{3}+2^{3}+2^{3}+\sum_{i=1}^{3}x_{i}^{3}$. If five of
the $x_{i}$’s are zero, then
$n=7^{3}+6^{3}+6^{3}+6^{3}+3^{3}+3^{3}+3^{3}+\sum_{i=1}^{2}x_{i}^{3}$. If six
of the $x_{i}$’s are zero, then
$n=7^{3}+6^{3}+6^{3}+6^{3}+4^{3}+2^{3}+2^{3}+1^{3}+x_{7}^{3}$. Hence, every
$n>1526$, has a representation as the sum of exactly $9$ positive cubes. It is
left to check for which values of $n\leq 1526$ are not the sum of $9$ cubes:

$\displaystyle\mathbf{B}_{9}^{3}=\\{1,2,\ldots,8\\}$
$\displaystyle\cup\\{9+\beta\mid\beta\in\mathbf{B}^{3}\\}$
$\displaystyle\cup\\{162,169,174,181,188,193,195,200,207,214,221,230,247,$
$\displaystyle\qquad
249,256,267,286,293,305,310,312,319,347,366,373,422,471\\}.$

For $j=10$, consider $n-1$ for $n>1526+1$. Because $n-1$ has a representation
as a sum of $9$ positive cubes, $n=\sum_{i=1}^{9}x_{i}^{3}+1^{3}$ is a
representation of $n$ as the sum of $10$ positive cubes. It is left to check
which values of $n\leq 1526+1$ are not the sum of $10$ cubes:

$\displaystyle\mathbf{B}_{10}^{3}=\\{1,2,\ldots,9\\}$
$\displaystyle\cup\\{10+\beta\mid\beta\in\mathbf{B}^{3}\\}$
$\displaystyle\cup\\{163,170,182,189,196,201,208,215,222,257,294,313,320,374\\}.$

For $j=11$, consider $n-1$ for $n>1526+2$. Because $n-1$ has a representation
as a sum of $11$ positive cubes, $n=\sum_{i=1}^{11}x_{i}^{3}+1^{3}$ is a
representation of $n$ as the sum of $12$ positive cubes. It is left to check
which values of $n\leq 1526+2$ are not the sum of $12$ cubes:

$\mathbf{B}_{11}^{3}=\\{1,2,\ldots,10\\}\cup\\{11+\beta\mid\beta\in\mathbf{B}^{3}\\}\cup\\{171,190,197,209,216,223,321\\}.$

For $j=12$, consider $n-1$ for $n>1526+3$. Because $n-1$ has a representation
as a sum of $11$ positive cubes, $n=\sum_{i=1}^{11}x_{i}^{3}+1^{3}$ is a
representation of $n$ as the sum of $12$ positive cubes. It is left to check
which values of $n\leq 1526+3$ are not the sum of $12$ cubes:

$\mathbf{B}_{12}^{3}=\\{1,2,\ldots,11\\}\cup\\{12+\beta\mid\beta\in\mathbf{B}^{3}\\}\cup\\{198,217,224\\}.$

For $j=13$, consider $n-1$ for $n>1526+4$. Because $n-1$ has a representation
as a sum of $12$ positive cubes, $n=\sum_{i=1}^{12}x_{i}^{3}+1^{3}$ is a
representation of $n$ as the sum of $13$ positive cubes. It is left to check
which values of $n\leq 1526+4$ are not the sum of $13$ cubes:

$\mathbf{B}_{13}^{3}=\\{1,2,\ldots,12\\}\cup\\{13+\beta\mid\beta\in\mathbf{B}^{3}\\}\cup\\{225\\}.$

For $j=14$, consider $n-1$ for $n>1526+5$. Because $n-1$ has a representation
as a sum of $13$ positive cubes, $n=\sum_{i=1}^{13}x_{i}^{3}+1^{3}$ is a
representation of $n$ as the sum of $14$ positive cubes. It is left to check
which values of $n<1526+5$ are not the sum of $14$ cubes:

$\mathbf{B}_{14}^{3}=\\{1,2,\ldots,13\\}\cup\\{14+\beta\mid\beta\in\mathbf{B}^{3}\\}.$

For $j=15$, consider $n-1$ for $n>1526+6$. Because $n-1$ has a representation
as a sum of $14$ positive cubes, $n=\sum_{i=1}^{14}x_{i}^{3}+1^{3}$ is a
representation of $n$ as the sum of $15$ positive cubes. It is left to check
which values of $n\leq 1526+6$ are not the sum of $15$ cubes:

$\mathbf{B}_{15}^{3}=\\{1,2,\ldots,14\\}\cup\\{15+\beta\mid\beta\in\mathbf{B}^{3}\\}.$

Note here that
$\mathbf{B}_{15}^{3}=\\{\beta+1\mid\beta\in\mathbf{B}_{14}^{3}\\}\cup\\{1\\}$.
The same processes outlined for $j=15$ can be repeated to show that for
$15\leq j\leq 22$,
$\mathbf{B}_{j+1}^{3}=\\{\beta-1\mid\beta\in\mathbf{B}_{j}^{3}\\}\cup\\{1\\}$.
Thus, for $j=21$, consider $n-1$ for $n>1526+12$. Because $n-1$ has a
representation as a sum of $20$ positive cubes,
$n=\sum_{i=1}^{20}x_{i}^{3}+1^{3}$ is a representation of $n$ as the sum of
$21$ positive cubes. It is left to check which values of $n\leq 1526+11$ are
not the sum of $21$ cubes:

$\mathbf{B}_{21}^{3}=\\{1,2,\ldots,20\\}\cup\\{21+\beta\mid\beta\in\mathbf{B}^{3}\\}.$

Finally, for $j=22$, consider $n-1$ for $n>1526+13$. Because $n-1$ has a
representation as a sum of $14$ positive cubes,
$n=\sum_{i=1}^{21}x_{i}^{3}+1^{3}$ is a representation of $n$ as the sum of
$22$ positive cubes. It is left to check which values of $n\leq 1526+13$ are
not the sum of $22$ cubes:

$\mathbf{B}_{22}^{3}=\\{1,2,\ldots,21\\}\cup\\{22+\beta\mid\beta\in\mathbf{B}^{3}\\}.$

It is clear that no element in the set $\\{1,2,\ldots,21\\}$ is the sum of
$22$ positive cubes, so consider elements in the set
$\\{22+\beta\mid\beta\in\mathbf{B}^{3}\\}$. If any element had a
representation as the sum of $22$ positive cubes, then the smallest term in
that representation would be $1^{3}$ since $22\cdot
2^{3}=176>171=22+\max{\\{\beta\in\mathbf{B}^{3}\\}}=22+149$. Hence, if any
element in the set $\\{22+\beta\mid\beta\in\mathbf{B}^{3}\\}$ is a sum of $22$
positive cubes, then $21+\beta=22+\beta-1$ for $\beta\in\mathbf{B}^{3}$ would
be the sum of $21$ positive cubes, contrary to what has already been shown.

In general, suppose for a given $k\geq 22$, every positive integer is the sum
of exactly $j$ positive cubes except for $1,2,\ldots,j-1$ and all $j+\beta$
for $\beta\in\mathbf{B}^{3}$. Then the integers $1,2,\ldots,j$ cannot be
represented as the sum of exactly $j+1$ positive cubes. If
$n-1=\sum_{i=1}^{j}x_{i}^{3}$ where $x_{j}\geq\ldots\geq x_{1}>0$, then
$n=\sum_{i-1}^{j+1}x_{i}^{3}$ where $x_{j+1}=1$. It remains to consider the
integers $(j+1)+\beta$ for $\beta\in\mathbf{B}^{3}$. Note again that for
$j\geq 22$, the smallest element in each representation is $1^{3}$ since
$(j+1)\cdot 2^{3}>(j+1)+149$. Thus if $(j+1)+\beta$ is the sum of exactly
$k+1$ positive cubes, then $j+\beta$ is the sum of exactly $j$ positive cubes,
contradicting the inductive hypothesis. ∎

The theorem established Siksek [59], that $G(3)\leq 7$, is the current best
result, but there are conjectures that $G(3)$ is much smaller. Romani [58]
conjectures that every integer greater than $1290740$ is the sum of at most
five positive cubes, and Deshouillers, Hennecart, & Landreau [24] conjecture
that every integer greater than $7373170279850$ is the sum of at most four
positive cubes. If this result holds, this same proof technique could be
improved, and the sets $\overline{\mathbf{B}_{8}^{3}}$ and
$\overline{\mathbf{B}_{7}^{3}}$ could be determined. Further results will
require other techniques.

##  6\. Proof of Theorem 3.8

Observe that $12785$ has the marvelous property that it is the sum of 15, 16,
17, …, 31 positive fourth powers.

$\displaystyle 12785$
$\displaystyle=8^{4}+8^{4}+8^{4}+4^{4}+3^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}$
$\displaystyle=10^{4}+6^{4}+6^{4}++2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+1^{4}$
$\displaystyle=9^{4}+7^{4}+7^{4}+5^{4}+5^{4}+3^{4}+3^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}$
$\displaystyle\ \ \vdots$
$\displaystyle=10^{4}+6^{4}+6^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+1^{4}+1^{4}+1^{4}$
$\displaystyle\ \ \
+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}$

The complete list of representations of $12785$ is listed in the Appendix.
Recall Lemma 3.7: that every integer greater than $13792$ can be expressed as
the sum of $16$ positive fourth powers. Let $n>12785+13792=26577$ and consider
when $j=16$. It follows that $n-12785=\sum_{i=1}^{16}x_{i}^{4}$ where
$x_{16}\geq\ldots\geq x_{1}\geq 0$. If all $x_{i}$’s are positive, then

$n=8^{4}+8^{4}+8^{4}+4^{4}+3^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+\sum_{i=1}^{16}x_{i}^{4}.$

If one of the $x_{i}$’s is zero, then

$n=10^{4}+6^{4}+6^{4}++2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+1^{4}+\sum_{i=1}^{15}x_{i}^{4}.$

If two of the $x_{i}$’s are zero, then

$n=9^{4}+7^{4}+7^{4}+5^{4}+5^{4}+3^{4}+3^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+\sum_{i=1}^{14}x_{i}^{4}$

This continues until, if 15 of the $x_{i}$’s are zero, then

$\displaystyle n$
$\displaystyle=10^{4}+6^{4}+6^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+2^{4}+1^{4}+1^{4}$
$\displaystyle\ \ \
+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+1^{4}+x_{1}^{4}.$

Hence, every $n>26577$ has a representation as the sum of exactly $31$
positive fourth powers. It is left to check for which values of $n\leq 26577$
are not the sum of $31$ positive fourth powers:

$\mathbf{B}_{31}^{4}=\\{1,2,\ldots,30\\}\cup\\{31+\beta\mid\beta\in\mathbf{B}^{4}\\}.$

For $j=32$, consider $n-1$ for $n>26577+1$. Because $n-1$ has a representation
as a sum of $31$ positive fourth powers, then
$n=\sum_{i=1}^{31}x_{i}^{4}+1^{4}$ is a representation of $n$ as the sum of
$32$ positive fourth powers. It is left to check for which values of
$n<26577+1$ are not the sum of $32$ positive fourth powers:

$\mathbf{B}_{32}^{4}=\\{1,2,\ldots,31\\}\cup\\{32+\beta\mid\beta\in\mathbf{B}^{4}\\}$

For $j=33$, consider $n-1$ for $n>26577+2$. Because $n-1$ has a representation
as a sum of $32$ positive fourth powers, then
$n=\sum_{i=1}^{32}x_{i}^{4}+1^{4}$ is a representation of $n$ as the sum of
$33$ positive fourth powers. It is left to check for which values of
$n<26577+2$ are not the sum of $33$ positive fourth powers:

$\mathbf{B}_{33}^{4}=\\{1,2,\ldots,31\\}\cup\\{32+\beta\mid\beta\in\mathbf{B}^{4}\\}$

This same process can be repeated to show that for $33\leq j\leq 920$, the set

$\mathbf{B}_{j+1}^{4}=\\{\beta-1\mid\beta\in\mathbf{B}_{j}^{4}\cup\\{1\\}.$

For $j=920$, consider $n-1$ for $n>26577+889$. Because $n-1$ has a
representation as a sum of $919$ positive fourth powers,
$n=\sum_{i=1}^{919}x_{i}^{4}+1^{4}$ is a representation of $n$ as the sum of
$920$ positive fourth powers. It is left to check for which values of
$n<26577+889$ are not the sum of $920$ positive fourth powers:

$\mathbf{B}_{920}^{4}=\\{1,2,\ldots,919\\}\cup\\{920+\beta\mid\beta\in\mathbf{B}^{4}\\}$

It is clear that no element in the set $\\{1,2,\ldots,919\\}$ is the sum of
$920$ positive fourth powers. So consider elements in the set
$\\{920+\beta\mid\beta\in\mathbf{B}^{4}\\}$. If any element in this set had a
representation as the sum of $920$ positive fourth powers, then the smallest
term in that representation would be $1^{4}$ since
$14720=920+2^{4}>920+13792=14712$. Hence, if any element in the set
$\\{920+\beta\mid\beta\in\mathbf{B}^{4}\\}$ is a sum of $920$ positive fourth
powers, then $919+\beta=920+\beta-1$ for $\beta\in\mathbf{B}^{4}$ would be the
sum of $919$ positive fourth powers, contrary to what has already been shown.

In general, suppose for a given $j>920$ that every positive integer $n$ is the
sum of exactly $j$ positive fourth powers, provided $n\neq 1,2,\ldots,j-1$ and
$n\neq j+\beta$ for $\beta\in\mathbf{B}^{4}$. It is clear that the integers
$1,2,\ldots,j-1$ cannot be expressed as the sum of $j$ positive fourth powers
- so consider positive integers $(j+1)+\beta$ for $\beta\in\mathbf{B}^{4}$.
Note again that for $j\geq 920$, the smallest term in each representation is
$1^{4}$ since $(j+1)\cdot 2^{4}>(j+1)+13792$. Thus if $(j+1)+\beta$ is the sum
of exactly $j+1$ positive fourth powers, then $j+\beta$ is the sum of exactly
$j$ positive fourth powers, contradicting the inductive hypothesis. ∎

##  7\. Proof of Theorems 3.14, & 3.15

###### Proof.

Observe that $39512917$ has the marvelous property that it is the sum of 6, 7,
8, …, 44 positive fifth powers.

$\displaystyle 39512917$
$\displaystyle=27^{5}+26^{5}+25^{5}+17^{5}+16^{5}+16^{5}$
$\displaystyle=28^{5}+27^{5}+23^{5}+15^{5}+14^{5}+11^{5}+9^{5}$
$\displaystyle=30^{5}+27^{5}+15^{5}+10^{5}+5^{5}+4^{5}+3^{5}+3^{5}$
$\displaystyle\ \vdots$
$\displaystyle=16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}$
$\displaystyle+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}$
$\displaystyle+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+16^{5}+15^{5}+15^{5}+15^{5}+15^{5}+14^{5}+14^{5}$
$\displaystyle+13^{5}+12^{5}+11^{5}+6^{5}+6^{5}+2^{5}+1$

The complete list of representations of $39512917$ is listed in the Appendix.
Let $n>39512917+N$ and consider when $j=23$. By Lemma 3.34,
$n-39512917=\sum_{i=1}^{17}x_{i}^{5}$ where $x_{17}\geq\ldots\geq x_{1}\geq
0$. If all $x_{i}$’s are positive, then

$n=27^{5}+26^{5}+25^{5}+17^{5}+16^{5}+16^{5}+\sum_{i=1}^{17}x_{i}^{5}.$

If one of the $x_{i}$’s is zero, then

$n=28^{5}+27^{5}+23^{5}+15^{5}+14^{5}+11^{5}+9^{5}+\sum_{i=1}^{16}x_{i}^{5}.$

If two of the $x_{i}$’s are zero, then

$n=30^{5}+27^{5}+15^{5}+10^{5}+5^{5}+4^{5}+3^{5}+3^{5}+\sum_{i=1}^{15}x_{i}^{5}.$

This continues until, if 16 of the $x_{i}$’s are zero, then

$\displaystyle n$
$\displaystyle=19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+19^{5}+18^{5}+18^{5}$
$\displaystyle\ \
+18^{5}+17^{5}+17^{5}+14^{5}+13^{5}+13^{5}+6^{5}+5^{5}+x_{1}^{5}.$

Assuming Conjecture 3.13, that the largest positive integer not expressible as
a sum of $17$ non-negative fifth powers is $87918$, then
$39512917+N=39600835$. In this case, every $n>39600835$ has a representation
as the sum of exactly $23$ positive fifth powers. It is left to check for
which values of $n\leq 39600835$ are not the sum of $23$ positive fifth
powers:

$\mathbf{B}_{23}^{5}=\\{1,2,\ldots,22\\}\cup\\{23+\beta\mid\beta\in\mathbf{B}^{5}\\}\cup\overline{\mathbf{B}_{23}^{5}}.$

The complete set of $\overline{\mathbf{B}_{23}^{5}}$ is listed in the
Appendix, provided Conjecture 3.13 holds. For $j=24$, consider $n-1$ for
$n>39600835+1$. Because $n-1$ has a representation as a sum of $23$ positive
fifth powers, then $n=\sum_{i=1}^{23}x_{i}^{5}+1^{5}$ is a representation of
$n$ as the sum of $24$ positive fifth powers. It is left to check for which
values of $n<39600835+1$ are not the sum of $24$ positive fifth powers:

$\mathbf{B}_{24}^{5}=\\{1,2,\ldots,23\\}\cup\\{24+\beta\mid\beta\in\mathbf{B}^{5}\\}\cup\overline{\mathbf{B}_{24}^{5}}.$

The complete set $\overline{\mathbf{B}_{24}^{5}}$ is listed in the Appendix,
conditional on Conjecture 3.13. For $j=25$, consider $n-1$ for $n>39600835+2$.
Because $n-1$ has a representation as a sum of $24$ positive fifth powers,
then $n=\sum_{i=1}^{24}x_{i}^{5}+1^{5}$ is a representation of $n$ as the sum
of $25$ positive fifth powers. It is left to check for which values of
$n<39600835+2$ are not the sum of $25$ positive fifth powers:

$\mathbf{B}_{25}^{5}=\\{1,2,\ldots,24\\}\cup\\{25+\beta\mid\beta\in\mathbf{B}^{5}\\}\cup\overline{\mathbf{B}_{25}^{5}}$

The complete set $\overline{\mathbf{B}_{25}^{5}}$ is listed in the Appendix,
again conditional on Conjecture 3.13. This continues inductively, where each
$\overline{\mathbf{B}_{j}^{5}}$ is nonempty (and listed for completeness in
the Appendix), until $j=57$. For $j=57$, consider $n-1$ for $n>39600835+34$.
Because $n-1$ has a representation as a sum of $56$ positive fifth powers,
then $n=\sum_{i=1}^{56}x_{i}^{5}+1^{5}$ is a representation of $n$ as the sum
of $57$ positive fifth powers. It is left to check for which values of
$n<39600835+32$ are not the sum of $57$ positive fifth powers:

$\mathbf{B}_{57}^{5}=\\{1,2,\ldots,56\\}\cup\\{57+\beta\mid\beta\in\mathbf{B}^{5}\\}\cup\\{9384\\}.$

Note that $\overline{\mathbf{B}_{57}^{5}}=\emptyset$. And, for $j=58$,
consider $n-1$ for $n>39600835+33$. Because $n-1$ has a representation as a
sum of $57$ positive fifth powers, then $n=\sum_{i=1}^{58}x_{i}^{5}+1^{5}$ is
a representation of $n$ as the sum of $58$ positive fifth powers. It is left
to check for which values of $n<39600835+32$ are not the sum of $58$ positive
fifth powers:

$\mathbf{B}_{58}^{5}=\\{1,2,\ldots,57\\}\cup\\{58+\beta\mid\beta\in\mathbf{B}^{5}\\}.$

Note that
$\mathbf{B}_{58}^{5}=\\{\beta+1\mid\beta\in\mathbf{B}_{57}^{5}\\}\cup\\{1\\}$.
The largest value in $\mathbf{B}_{57}^{5}$ is $6318$, so in Theorem 4.18,
$m=6318,k=5$, and $j=57$. The first condition is satisfied, and the second
condition evaluates to

$\left\lfloor\sqrt[5]{6261+\frac{6261}{31}}\right\rfloor=\left\lfloor\sqrt[5]{6318}\right\rfloor.$

∎

Because Vaughan & Wooley [78] showed that $G(5)\leq 17$ for sufficiently large
positive integers, let $n$ be sufficiently large. From here, the procedure
could follow as in the proofs for the case of cubes and the case of fourths,
and the proofs of Theorems 3.14 and 3.15 follow. Knowing that the proof works
for sufficiently large $n$ indicates that there must be a largest such $n$ for
which the statement of the Theorem fails. Conjecture 3.13 predicts that the
largest such positive integer is $87918$. The proof proceeds using this
conjectured value. If the conjecture does not hold, then a substitution of the
correct smallest such value is easily performed without any loss of validity
to the proof.

Note that $39512917$ works in the proof of Theorems 3.14, & 3.15 because it is
small enough for modest computers to find consecutive expressions as the sum
of $6,7,8,\ldots,44$ positive fifth powers, but there are larger numbers that
would improve the results of Theorems 3.14, & 3.15. For example, the smallest
positive integer that is the sum of $2$ and $3$ positive fifth powers is
known:

$563661204304422162432=14132^{5}+220^{5}=14068^{5}+6237^{5}+5027^{5}.$

How far can this technique work for fifths?

There are almost certainly not positive integers that are the sum of
$2,3,4,\ldots,40$ positive fifth powers, considering a density argument, but
perhaps such a number exists yet is inaccessibly large. If so, this would
allow proof of which numbers are not the sum of $j\geq 19$ positive fifth
powers. This is the theoretical limit that this proof technique will allow for
positive fifth powers without further improvements in Waring’s problem. One
possible improvement that requires more investigation is the following:

######  Conjecture 7.1.

Every positive integer greater than $77529941$ is the sum of exactly $10$
positive fifth powers.

##  8\. Proof of Theorems 3.20 & 3.21

###### Proof.

Observe that $237071474$ has the marvelous property that it is the sum of
$9,10,11,\ldots,69$ positive sixth powers.

$\displaystyle 237071474$
$\displaystyle=22^{6}+21^{6}+14^{6}+14^{6}+14^{6}+14^{6}+14^{6}+8^{6}+5^{6}$
$\displaystyle=22^{6}+19^{6}+18^{6}+16^{6}+16^{6}+13^{6}+12^{6}+10^{6}+8^{6}+4^{6}$
$\displaystyle=20^{6}+20^{6}+20^{6}+16^{6}+16^{6}+14^{6}+12^{6}+10^{6}+3^{6}+3^{6}+2^{6}$
$\displaystyle\ \ \vdots$
$\displaystyle=13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}$
$\displaystyle\ \ \
+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}$
$\displaystyle\ \ \
+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}+13^{6}$
$\displaystyle\ \ \
+12^{6}+12^{6}+12^{6}+12^{6}+12^{6}+12^{6}+12^{6}+12^{6}+12^{6}+12^{6}+12^{6}+12^{6}+11^{6}$
$\displaystyle\ \ \
+11^{6}+11^{6}+10^{6}+10^{6}+10^{6}+10^{6}+10^{6}+10^{6}+9^{6}+9^{6}+8^{6}+7^{6}+7^{6}+7^{6}$
$\displaystyle\ \ \ +3^{6}+3^{6}+1^{6}.$

The complete list of representations of $237071474$ can be found in the
Appendix. Let $N$ be the largest positive integer not expressible as the sum
of at most $24$ sixth powers, and $n>237071474+N$. Consider when $j=33$. By
Lemma 3.17, $n-237071474=\sum_{i=1}^{24}x_{i}^{6}$ where $x_{24}\geq\ldots\geq
x_{1}\geq 0$. If all $x_{i}$’s are positive, then

$n=22^{6}+21^{6}+14^{6}+14^{6}+14^{6}+14^{6}+14^{6}+8^{6}+5^{6}+\sum_{i=1}^{24}x_{i}^{6}.$

If one of the $x_{i}$’s is zero, then

$n=22^{6}+19^{6}+18^{6}+16^{6}+16^{6}+13^{6}+12^{6}+10^{6}+8^{6}+4^{6}+\sum_{i=1}^{23}x_{i}^{6}.$

If two of the $x_{i}$’s are zero, then

$n=20^{6}+20^{6}+20^{6}+16^{6}+16^{6}+14^{6}+12^{6}+10^{6}+3^{6}+3^{6}+2^{6}+\sum_{i=1}^{22}x_{i}^{6}.$

This continues until, if 23 of the $x_{i}$’s are zero, then

$\displaystyle n$
$\displaystyle=15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}+15^{6}$
$\displaystyle+15^{6}+15^{6}+15^{6}+15^{6}+14^{6}+13^{6}+11^{6}+11^{6}+11^{6}+11^{6}+9^{6}+9^{6}+6^{6}+6^{6}+6^{6}+4^{6}$
$\displaystyle+2^{6}+x_{1}^{6}.$

Then, every $n>237071474+N$ has a representation as the sum of exactly $33$
positive sixth powers. Therefore,

$\mathbf{B}_{33}^{6}=\\{1,2,\ldots,32\\}\cup\\{33+\beta\mid\beta\in\mathbf{B}^{6}\\}\cup\overline{\mathbf{B}_{33}^{6}},$

where $\overline{\mathbf{B}_{33}^{6}}$ is a finite set. A conjectured complete
set of $\overline{\mathbf{B}_{33}^{6}}$ is listed in the Appendix.

To get a numeric upper bound for $\overline{\mathbf{B}_{j}^{6}}$, it is
necessary to resort to the general result of Waring’s problem. Since $g(6)=73$
according to Pillai [56], every positive integer is the sum of either
$1,2,\ldots$, or $73$ positive sixth powers. For an arbitrary positive integer
$n>237071474$, let $n-237071474=\sum_{i=0}^{j}x_{i}^{6}$, where $n$ can be
represented as a sum of $j$ sixth powers. Then,
$237071474=\sum_{i=1}^{82-j}y_{i}^{6}$ for some sequence of $y_{i}$ depending
on $j$. Hence,
$n=(n-237071474)+237071474=\sum_{i=1}^{j}x_{i}^{6}+\sum_{i=1}^{82-j}y_{i}^{6}$,
which is a sum of $82$ positive sixth powers. It is left to check for which
values of $n<237071474$ are not the sum of $82$ positive sixth powers:

$\mathbf{B}_{82}^{6}=\\{1,2,\ldots,81\\}\cup\\{82+\beta\mid\beta\in\mathbf{B}^{6}\\}.$

Now, for $j=83$, consider $n-1$ for $n>237071474+1$. Because $n-1$ has a
representation as a sum of $82$ positive sixth powers, then
$n=\sum_{i=1}^{82}x_{i}^{6}+1^{3}$ is a representation of $n$ as the sum of
$83$ positive sixth powers. It is left to check for which values of
$n<237071474+1$ are not the sum of $83$ positive sixth powers:

$\mathbf{B}_{83}^{6}=\\{1,2,\ldots,82\\}\cup\\{83+\beta\mid\beta\in\mathbf{B}^{6}\\}.$

Note that
$\mathbf{B}_{83}^{6}=\\{\beta+1\mid\beta\in\mathbf{B}_{82}^{6}\\}\cup\\{1\\}$.
The largest value in $\mathbf{B}_{82}^{6}$ is $715826$, so in Theorem 4.18,
$m=715826,k=6$, and $j=82$. The first condition is satisfied, and the second
condition evaluates to

$\left\lfloor\sqrt[6]{715744+\frac{715744}{63}}\right\rfloor=\left\lfloor\sqrt[6]{715826}\right\rfloor.$

∎

Because Vaughan & Wooley [78] showed that $G(6)\leq 24$; every sufficiently
large positive integer must be the sum of at most $24$ positive sixth powers.
Conjecture 3.18 predicts that the largest such positive integer is $1414564$.
Hence, there must be a largest such $n$ for which the statement of Theorem
3.20 fails. The proof proceeds using this conjectured value. If the conjecture
does not hold, then a substitution of the correct smallest such value is
easily performed without any loss of validity to the proof.

Note that $237071474$ works in the proof of Theorem 3.20 because it is small
enough for modest computers but there are larger numbers that would improve
the results. How far can this technique work for positive sixth powers? There
are almost certainly not positive integers that are the sum of
$2,3,4,\ldots,69$ positive sixth powers, considering a density argument, but
perhaps such a number exists yet is inaccessibly large. If so, this would
allow proof of which numbers are not the sum of $j\geq 80$ positive sixth
powers. This is the theoretical limit that this proof technique will allow for
positive sixth powers without further improvements in Waring’s problem. One
possible improvement that requires more investigation is the following:

######  Conjecture 8.1.

Every positive integer greater than $229182966$ is the sum of $18$ positive
sixth powers.

##  9\. Proof of Theorems 3.26 & 3.25

###### Proof.

Observe that $340425660$ has the marvelous property that it is the sum of
$18,19,20,\ldots,32$ (and many more!) positive seventh powers.

$\displaystyle 340425660$
$\displaystyle=15^{7}+13^{7}+13^{7}+12^{7}+8^{7}+8^{7}+8^{7}+7^{7}+6^{7}+6^{7}+6^{7}+6^{7}+3^{7}$
$\displaystyle\ \ \ +2^{7}+2^{7}+2^{7}+2^{7}+1^{7}$
$\displaystyle=15^{7}+12^{7}+11^{7}+11^{7}+11^{7}+11^{7}+11^{7}+10^{7}+9^{7}+9^{7}+9^{7}+9^{7}+9^{7}$
$\displaystyle\ \ \ +8^{7}+6^{7}+3^{7}+3^{7}+3^{7}+2^{7}$
$\displaystyle=15^{7}+14^{7}+11^{7}+11^{7}+11^{7}+9^{7}+6^{7}+6^{7}+6^{7}+4^{7}+4^{7}+4^{7}+4^{7}+3^{7}+2^{7}$
$\displaystyle\ \ \ +2^{7}+2^{7}+2^{7}+2^{7}+2^{7}$ $\displaystyle\ \ \vdots$
$\displaystyle=12^{7}+12^{7}+12^{7}+12^{7}+12^{7}+11^{7}+11^{7}+11^{7}+11^{7}+11^{7}+11^{7}+10^{7}+10^{7}$
$\displaystyle\ \ \
+9^{7}+9^{7}+8^{7}+8^{7}+8^{7}+8^{7}+8^{7}+8^{7}+8^{7}+5^{7}+4^{7}+3^{7}+2^{7}+2^{7}$
$\displaystyle\ \ \ +2^{7}+2^{7}+2^{7}+2^{7}+2^{7}.$

The complete list of representations of $340425660$ can be found in the
Appendix. Let $N$ be the largest positive integer not expressible as the sum
of at most $33$ positive seventh powers. Let $n>340425660+N$ and consider when
$j=51$. By Conjecture 3.23, $n-340425660=\sum_{i=1}^{33}x_{i}^{7}$ where
$x_{51}\geq\ldots\geq x_{1}\geq 0$. If all $x_{i}$’s are positive, then

$\displaystyle n=$ $\displaystyle
15^{7}+13^{7}+13^{7}+12^{7}+8^{7}+8^{7}+8^{7}+7^{7}+6^{7}+6^{7}+6^{7}+6^{7}+3^{7}$
$\displaystyle\ \ \ +2^{7}+2^{7}+2^{7}+2^{7}+1^{7}+\sum_{i=1}^{33}x_{i}^{7}.$

If one of the $x_{i}$’s is zero, then

$\displaystyle n=$ $\displaystyle
15^{7}+12^{7}+11^{7}+11^{7}+11^{7}+11^{7}+11^{7}+10^{7}+9^{7}+9^{7}+9^{7}+9^{7}+9^{7}$
$\displaystyle\ \ \
+8^{7}+6^{7}+3^{7}+3^{7}+3^{7}+2^{7}+\sum_{i=1}^{32}x_{i}^{7}.$

This continues until, if $32$ of the $x_{i}$’s are zero, then

$\displaystyle n$
$\displaystyle=12^{7}+12^{7}+12^{7}+12^{7}+12^{7}+11^{7}+11^{7}+11^{7}+11^{7}+11^{7}+11^{7}+10^{7}+10^{7}$
$\displaystyle\ \ \
+9^{7}+9^{7}+8^{7}+8^{7}+8^{7}+8^{7}+8^{7}+8^{7}+8^{7}+5^{7}+4^{7}+3^{7}+2^{7}+2^{7}$
$\displaystyle\ \ \ +2^{7}+2^{7}+2^{7}+2^{7}+2^{7}+x_{1}^{7}.$

Assuming Conjecture 3.23, that the largest positive integer not expressible as
a sum of $33$ non-negative seventh powers is $45987024$, then
$340425660+N=386412684$. In this case, every $n>386412684$ has a
representation as the sum of exactly $51$ positive seventh powers. It is left
to check for which values of $n\leq 386412684$ are not the sum of $51$
positive seventh powers:

$\mathbf{B}_{51}^{7}=\\{1,2,\ldots,50\\}\cup\\{51+\beta\mid\beta\in\mathbf{B}^{7}\\}\cup\overline{\mathbf{B}_{51}^{7}}.$

The complete set of $\overline{\mathbf{B}_{51}^{7}}$ is listed in the
Appendix, provided Conjecture 3.23 holds. For $j=52$, consider $n-1$ for
$n>386412684+1$. Because $n-1$ has a representation as a sum of $51$ positive
seventh powers, then $n=\sum_{i=1}^{51}x_{i}^{7}+1^{7}$ is a representation of
$n$ as the sum of $52$ positive seventh powers. It is left to check for which
values of $n<386412684+1$ are not the sum of $52$ positive seventh powers:

$\mathbf{B}_{52}^{7}=\\{1,2,\ldots,51\\}\cup\\{52+\beta\mid\beta\in\mathbf{B}^{7}\\}\cup\overline{\mathbf{B}_{52}^{7}}.$

The complete set $\overline{\mathbf{B}_{52}^{7}}$ is listed in the Appendix,
provided Conjecture 3.23 holds. For $j=53$, consider $n-1$ for
$n>386412684+2$. Because $n-1$ has a representation as a sum of $52$ positive
seventh powers, then $n=\sum_{i=1}^{52}x_{i}^{7}+1^{7}$ is a representation of
$n$ as the sum of $53$ positive seventh powers. It is left to check for which
values of $n<386412684+2$ are not the sum of $53$ positive seventh powers:

$\mathbf{B}_{53}^{7}=\\{1,2,\ldots,52\\}\cup\\{53+\beta\mid\beta\in\mathbf{B}^{7}\\}\cup\overline{\mathbf{B}_{53}^{7}}$

The complete set $\overline{\mathbf{B}_{53}^{7}}$ is listed in the Appendix,
again provided Conjecture 3.23 holds. This continues inductively, where each
$\overline{\mathbf{B}_{j}^{7}}$ is nonempty, until $j=246$. For $j=246$,
consider $n-1$ for $n>386412684+194$. Because $n-1$ has a representation as a
sum of $245$ positive seventh powers, then $n=\sum_{i=1}^{245}x_{i}^{7}+1^{7}$
is a representation of $n$ as the sum of $246$ positive seventh powers. It is
left to check for which values of $n<386412684+194$ are not the sum of $246$
positive seventh powers:

$\mathbf{B}_{246}^{7}=\\{1,2,\ldots,245\\}\cup\\{246+\beta\mid\beta\in\mathbf{B}^{7}\\}.$

Note that $\overline{\mathbf{B}_{246}^{7}}=\emptyset$. And, for $j=247$,
consider $n-1$ for $n>386412684+195$. Because $n-1$ has a representation as a
sum of $246$ positive seventh powers, then $n=\sum_{i=1}^{246}x_{i}^{7}+1^{7}$
is a representation of $n$ as the sum of $247$ positive seventh powers. It is
left to check for which values of $n<386412684+195$ are not the sum of $247$
positive seventh powers:

$\mathbf{B}_{247}^{7}=\\{1,2,\ldots,246\\}\cup\\{247+\beta\mid\beta\in\mathbf{B}^{7}\\}.$

Note that
$\mathbf{B}_{247}^{7}=\\{\beta+1\mid\beta\in\mathbf{B}_{246}^{7}\\}\cup\\{1\\}$.
The largest value in $\mathbf{B}_{246}^{7}$ is $249077$. Considering the
conditions in Theorem 4.18, $m=249077,k=7$, and $j=246$, the first condition
is satisfied, and the second condition evaluates to

$\left\lfloor\sqrt[7]{249077+\frac{249077}{2^{7}-1}}\right\rfloor=\left\lfloor\sqrt[7]{249323}\right\rfloor.$

∎

Because Vaughan & Wooley [78] showed that $G(7)\leq 33$ for sufficiently large
positive integers, the proof proceeds using the conjectured largest value that
cannot be expressed as the sum of $33$ positive seventh powers. If Conjecture
3.23 does not hold, then a substitution of the correct smallest such value is
easily performed without any loss of validity to the proof.

Note that $386412684$ works in the proof of Theorems 3.26, & 3.25 because it
is small enough for modest computers to find consecutive expressions as the
sum of $18,19,\ldots,32$ positive seventh powers, but there are (almost
certainly) larger numbers that would improve these results. How far can this
technique work for positive seventh powers?

There are almost certainly not positive integers that are the sum of
$2,3,4,\ldots,32$ positive seventh powers, considering a density argument, but
perhaps such a number exists yet is inaccessibly large. If so, this would
allow proof of which numbers are not the sum of $j\geq 35$ positive seventh
powers. This is the theoretical limit that this proof technique will allow for
positive seventh powers without further improvements in Waring’s problem. One
possible improvement that requires more investigation is the following:

######  Conjecture 9.1.

Every positive integer greater than $9930770$ is the sum of at most $33$
positive seventh powers.

######  Conjecture 9.2.

Every positive integer greater than $317476671$ is the sum of exactly $25$
positive seventh powers.

##  10\. Proofs 8ths

###### Proof.

Observe that $967214052$ has the marvelous property that it is the sum of
$22,23,24,\ldots,63$ (and many more!) positive eighth powers.

$\displaystyle 286959055$
$\displaystyle=10^{8}+10^{8}+8^{8}+8^{8}+8^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+5^{8}+5^{8}+5^{8}+5^{8}+5^{8}$
$\displaystyle\ \ \ \ \ \ \ \ +4^{8}+3^{8}+3^{8}+3^{8}+2^{8}+1^{8}$
$\displaystyle=9^{8}+9^{8}+9^{8}+9^{8}+9^{8}+8^{8}+8^{8}+8^{8}+8^{8}+6^{8}+6^{8}+5^{8}+5^{8}+5^{8}+4^{8}+3^{8}$
$\displaystyle\ \ \ \ \ \ \ \ +3^{8}+3^{8}+2^{8}+1^{8}+1^{8}+1^{8}+1^{8}$
$\displaystyle=10^{8}+10^{8}+9^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+6^{8}+6^{8}+6^{8}+6^{8}+6^{8}+5^{8}$
$\displaystyle\ \ \ \ \ \ \ \
+5^{8}+4^{8}+4^{8}+3^{8}+3^{8}+1^{8}+1^{8}+1^{8}+1^{8}$ $\displaystyle\ \
\vdots$
$\displaystyle=8^{8}+8^{8}+8^{8}+8^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}$
$\displaystyle\ \ \ \ \ \ \
+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}$
$\displaystyle\ \ \ \ \ \ \
+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+6^{8}+6^{8}+6^{8}+5^{8}+5^{8}$
$\displaystyle\ \ \ \ \ \ \
+5^{8}+4^{8}+4^{8}+4^{8}+4^{8}+4^{8}+3^{8}+3^{8}+2^{8}+2^{8}+2^{8}+2^{8}+1^{8}+1^{8}+1^{8}$
$\displaystyle\ \ \ \ \ \ \ +1^{8}+1^{8}.$

The complete list of representations of $28695055$ are listed in the Appendix.
Let $N$ be the largest positive integer not expressible as the sum of at most
$42$ positive eighth powers. Let $n>286959055+N$ and consider when $j=64$.
Despite not knowing the largest number that is not the sum of at most $42$
positive eighth powers, Vaughan and Wooley [77] showed that there exists a
smallest such number; denote that smallest number by $v$. The proof will
continue in the same manner as the previous proofs, with $v$ substituted for a
specific number. Choose any $n>286959055+v$. Then,
$n-286959055=\sum_{i=1}^{42}x_{i}^{8}$ where $x_{42}\geq\ldots\geq x_{1}\geq
0$. If all $x_{i}$’s are positive, then

$\displaystyle n=$ $\displaystyle\
10^{8}+10^{8}+8^{8}+8^{8}+8^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+5^{8}+5^{8}+5^{8}+5^{8}+5^{8}+4^{8}$
$\displaystyle\ \ \ \ \
+3^{8}+3^{8}+3^{8}+2^{8}+1^{8}+\sum_{i=1}^{42}x_{i}^{8}.$

If one of the $x_{i}$’s is zero, then

$\displaystyle n=$ $\displaystyle\
9^{8}+9^{8}+9^{8}+9^{8}+9^{8}+8^{8}+8^{8}+8^{8}+8^{8}+6^{8}+6^{8}+5^{8}+5^{8}+5^{8}+4^{8}+3^{8}+3^{8}$
$\displaystyle\ \ \ \ \
+3^{8}+2^{8}+1^{8}+1^{8}+1^{8}+1^{8}+\sum_{i=1}^{41}x_{i}^{8}.$

This continues until, if $32$ of the $x_{i}$’s are zero, then

$\displaystyle n$
$\displaystyle=8^{8}+8^{8}+8^{8}+8^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}$
$\displaystyle\ \ \ \ \ \ \
+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}$
$\displaystyle\ \ \ \ \ \ \
+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+7^{8}+6^{8}+6^{8}+6^{8}+5^{8}+5^{8}$
$\displaystyle\ \ \ \ \ \ \
+5^{8}+4^{8}+4^{8}+4^{8}+4^{8}+4^{8}+3^{8}+3^{8}+2^{8}+2^{8}+2^{8}+2^{8}+1^{8}+1^{8}+1^{8}$
$\displaystyle\ \ \ \ \ \ \ +1^{8}+1^{8}+x_{1}^{8}.$

To obtain a set with a definite upper bound, consider sums of $69$ positive
eighth powers. Let $N$ be the largest integer that is not the sum of at most
$47$ eighth powers. Assuming Conjecture 3.23, that the largest positive
integer not expressible as a sum of $47$ positive eighth powers is
$904339959$, then $286959055+N=1191299014$. In this case, every $n>1191299014$
has a representation as the sum of exactly $69$ positive eighth powers. It is
left to check for which values of $n\leq 1191299014$ are not the sum of $69$
positive eighth powers:

$\mathbf{B}_{69}^{8}=\\{1,2,\ldots,68\\}\cup\\{69+\beta\mid\beta\in\mathbf{B}^{8}\cup\overline{\mathbf{B}_{69}^{8}}\\}.$

The complete set of $\overline{\mathbf{B}_{69}^{8}}$ is listed in the
Appendix, provided Conjecture 3.29 holds. For $j=70$, consider $n-1$ for
$n>1191299014+1$. Because $n-1$ has a representation as a sum of $69$ positive
eighth powers, then $n=\sum_{i=1}^{69}x_{i}^{8}+1^{8}$ is a representation of
$n$ as the sum of $70$ positive eighth powers. It is left to check for which
values of $n<1191299014+1$ are not the sum of $70$ positive eighth powers:

$\mathbf{B}_{70}^{8}=\\{1,2,\ldots,69\\}\cup\\{70+\beta\mid\beta\in\mathbf{B}^{8}\cup\overline{\mathbf{B}_{70}^{8}}\\}.$

The complete set $\overline{\mathbf{B}_{70}^{8}}$ is listed in the Appendix,
provided Conjecture 3.29 holds. For $j=71$, consider $n-1$ for
$n>1191299014+2$. Because $n-1$ has a representation as a sum of $70$ positive
eighth powers, then $n=\sum_{i=1}^{70}x_{i}^{8}+1^{8}$ is a representation of
$n$ as the sum of $71$ positive eighth powers. It is left to check for which
values of $n<1191299014+2$ are not the sum of $71$ positive eighth powers:

$\mathbf{B}_{71}^{8}=\\{1,2,\ldots,70\\}\cup\\{71+\beta\mid\beta\in\mathbf{B}^{8}\cup\overline{\mathbf{B}_{71}^{8}}\\}.$

The complete set $\overline{\mathbf{B}_{71}^{8}}$ is listed in the Appendix,
again provided Conjecture 3.29 holds. This continues inductively, where each
$\overline{\mathbf{B}_{j}^{8}}$ is nonempty, until $j=334$. For $j=334$,
consider $n-1$ for $n>1191299014+266$. Because $n-1$ has a representation as a
sum of $333$ positive eighth powers, then $n=\sum_{i=1}^{333}x_{i}^{8}+1^{8}$
is a representation of $n$ as the sum of $334$ positive eighth powers. It is
left to check for which values of $n<386412684+265$ are not the sum of $334$
positive eighth powers:

$\mathbf{B}_{334}^{8}=\\{1,2,\ldots,333\\}\cup\\{334+\beta\mid\beta\in\mathbf{B}^{8}\\}.$

Note that $\overline{\mathbf{B}_{334}^{8}}=\emptyset$. And, for $j=335$,
consider $n-1$ for $n>386412684+266$. Because $n-1$ has a representation as a
sum of $334$ positive eighth powers, then $n=\sum_{i=1}^{334}x_{i}^{8}+1^{8}$
is a representation of $n$ as the sum of $335$ positive eighth powers. It is
left to check for which values of $n<1191299014+266$ are not the sum of $335$
positive eighth powers:

$\mathbf{B}_{335}^{8}=\\{1,2,\ldots,334\\}\cup\\{334+\beta\mid\beta\in\mathbf{B}^{8}\\}.$

Note that
$\mathbf{B}_{335}^{8}=\\{\beta+1\mid\beta\in\mathbf{B}_{334}^{8}\\}\cup\\{1\\}$.
The largest value in $\mathbf{B}_{334}^{8}$ is $1890241$. Considering the
conditions in Theorem 4.18, $m=1890241,k=8$, and $j=334$, the first condition
is satisfied, and the second condition evaluates to

$\left\lfloor\sqrt[8]{1890241+\frac{1890241}{2^{8}-1}}\right\rfloor=\left\lfloor\sqrt[8]{1890488}\right\rfloor.$

∎

Because Vaughan & Wooley [78] showed that $G(8)\leq 42$ for sufficiently large
positive integers, the proof proceeds using the conjectured largest value that
cannot be expressed as the sum of $42$ positive eighth powers. If Conjecture
3.29 does not hold, then a substitution of the correct smallest such value is
easily performed without any loss of validity to the proof.

Note that $386412684$ works in the proof of Theorems 3.30, & 3.32 because it
is small enough for modest computers to find consecutive expressions as the
sum of $22,23,\ldots,63$ positive eighth powers, but there are (almost
certainly) larger numbers that would improve these results. How far can this
technique work for positive eighth powers?

There are almost certainly not positive integers that are the sum of
$2,3,4,\ldots,43$ positive eighth powers, considering a density argument, but
perhaps such a number exists yet is inaccessibly large. If so, this would
allow proof of which numbers are not the sum of $j\geq 44$ positive eighth
powers. This is the theoretical limit that this proof technique will allow for
positive eighth powers without further improvements in Waring’s problem. One
possible improvement that requires more investigation is the following.

######  Conjecture 10.1.

Every positive integer greater than $904339959$ is the sum of exactly $47$
positive eighth powers.

######  Conjecture 10.2.

Every positive integer greater than $858367748$ is the sum of at most $42$
positive eighth powers.

##  11\. Proofs 9ths

###### Proof.

Observe that $967214052$ has the marvelous property that it is the sum of
$67,68,69,\ldots,187$ (and many more!) positive ninth powers.

$\displaystyle 967214052$
$\displaystyle=8^{9}+8^{9}+8^{9}+8^{9}+8^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+6^{9}+6^{9}+6^{9}+6^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+2^{9}+2^{9}+2^{9}$
$\displaystyle\ \ \ \ \ \ \
+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}$
$\displaystyle\ \ \ \ \ \ \
+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+1^{9}$
$\displaystyle\ \ \ \ \ \ \ +1^{9}+1^{9}$
$\displaystyle=7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}$
$\displaystyle\ \ \ \ \ \ \
+7^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+4^{9}+4^{9}+4^{9}+4^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}$
$\displaystyle\ \ \ \ \ \ \ +3^{9}+3^{9}+2^{9}$
$\displaystyle=8^{9}+8^{9}+8^{9}+8^{9}+8^{9}+8^{9}+8^{9}+7^{9}+5^{9}+4^{9}+4^{9}+4^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}$
$\displaystyle\ \ \ \ \ \ \
+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}+2^{9}+2^{9}+2^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}$
$\displaystyle\ \ \ \ \ \ \
+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}$
$\displaystyle\ \ \ \ \ \ \
+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}$
$\displaystyle\ \ \ \ \ \ \ +1^{9}+1^{9}+1^{9}+1^{9}$ $\displaystyle\vdots$
$\displaystyle=6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+4^{9}+4^{9}+2^{9}+2^{9}+2^{9}+2^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}$
$\displaystyle\ \ \ \ \ \ \
+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}$

The complete list of representations of $982835352$ are listed in the
Appendix. Let $N$ be the largest positive integer not expressible as the sum
of at most $50$ positive ninth powers. Let $n>982835352+N$ and consider when
$j=117$. Despite not knowing the largest number that is not the sum of at most
$50$ positive ninth powers, we know from Vaughan and Wooley [77] that there
exists a smallest such number; denote that smallest number by $v$. The proof
will continue in the same manner as the previous proofs, with $v$ substituted
for a specific number. Choose any $n>982835352+v$. Then,
$n-982835352=\sum_{i=1}^{50}x_{i}^{9}$ where $x_{50}\geq\ldots\geq x_{1}\geq
0$. If all $x_{i}$’s are positive, then

$\displaystyle n=$ $\displaystyle
8^{9}+8^{9}+8^{9}+8^{9}+8^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+6^{9}+6^{9}+6^{9}+6^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+2^{9}+2^{9}+2^{9}$
$\displaystyle\ \ \ \ \ \ \
+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}$
$\displaystyle\ \ \ \ \ \ \
+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+2^{9}+1^{9}$
$\displaystyle\ \ \ \ \ \ \ +1^{9}+1^{9}+\sum_{i=1}^{50}x_{i}^{9}.$

If one of the $x_{i}$’s is zero, then

$\displaystyle n=$ $\displaystyle
7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}+7^{9}$
$\displaystyle\ \ \ \ \ \ \
+7^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+4^{9}+4^{9}+4^{9}+4^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}+3^{9}$
$\displaystyle\ \ \ \ \ \ \ +3^{9}+3^{9}+2^{9}\sum_{i=1}^{49}x_{i}^{9}.$

This continues until, if $49$ of the $x_{i}$’s are zero, then

$\displaystyle n$
$\displaystyle=6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}+6^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}$
$\displaystyle\ \ \ \ \ \ \
+5^{9}+5^{9}+5^{9}+5^{9}+5^{9}+4^{9}+4^{9}+2^{9}+2^{9}+2^{9}+2^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}$
$\displaystyle\ \ \ \ \ \ \
+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+1^{9}+x_{1}^{9}.$

To obtain a set with a definite upper bound, consider sums of $188$ positive
ninth powers. Let $N$ be the largest integer that is not the sum of at most
$121$ ninth powers. Assuming Conjecture 3.36, that the largest positive
integer not expressible as a sum of $121$ positive ninth powers is
$967214052$, then $982835352+N=1950049404$. In this case, every $n>1950049404$
has a representation as the sum of exactly $188$ positive ninth powers. It is
left to check for which values of $n\leq 1950049404$ are not the sum of $188$
positive ninth powers:

$\mathbf{B}_{188}^{9}=\\{1,2,\ldots,187\\}\cup\\{188+\beta\mid\beta\in\mathbf{B}^{9}\cup\overline{\mathbf{B}_{188}^{9}}\\}.$

The complete set of $\overline{\mathbf{B}_{188}^{9}}$ is listed in the
Appendix, provided Conjecture 3.36 holds. For $j=189$, consider $n-1$ for
$n>1950049404+1$. Because $n-1$ has a representation as a sum of $188$
positive ninth powers, then $n=\sum_{i=1}^{188}x_{i}^{9}+1^{9}$ is a
representation of $n$ as the sum of $189$ positive ninth powers. It is left to
check for which values of $n<1950049404+1$ are not the sum of $189$ positive
ninth powers:

$\mathbf{B}_{189}^{9}=\\{1,2,\ldots,188\\}\cup\\{189+\beta\mid\beta\in\mathbf{B}^{9}\cup\overline{\mathbf{B}_{189}^{9}}\\}.$

The complete set $\overline{\mathbf{B}_{189}^{9}}$ is listed in the Appendix,
provided Conjecture 3.36 holds. For $j=190$, consider $n-1$ for
$n>1950049404+2$. Because $n-1$ has a representation as a sum of $189$
positive ninth powers, then $n=\sum_{i=1}^{189}x_{i}^{9}+1^{9}$ is a
representation of $n$ as the sum of $190$ positive ninth powers. It is left to
check for which values of $n<1950049404+2$ are not the sum of $190$ positive
ninth powers:

$\mathbf{B}_{190}^{9}=\\{1,2,\ldots,189\\}\cup\\{190+\beta\mid\beta\in\mathbf{B}^{9}\cup\overline{\mathbf{B}_{190}^{9}}\\}.$

The complete set $\overline{\mathbf{B}_{190}^{9}}$ is listed in the Appendix,
again provided Conjecture 3.36 holds. This continues inductively, where each
$\overline{\mathbf{B}_{j}^{9}}$ is nonempty, until $j=717$. For $j=717$,
consider $n-1$ for $n>1950049404+529$. Because $n-1$ has a representation as a
sum of $716$ positive ninth powers, then $n=\sum_{i=1}^{716}x_{i}^{9}+1^{9}$
is a representation of $n$ as the sum of $717$ positive ninth powers. It is
left to check for which values of $n<1950049404+529$ are not the sum of $717$
positive ninth powers:

$\mathbf{B}_{717}^{9}=\\{1,2,\ldots,716\\}\cup\\{717+\beta\mid\beta\in\mathbf{B}^{9}\\}.$

Note that $\overline{\mathbf{B}_{717}^{9}}=\emptyset$. And, for $j=718$,
consider $n-1$ for $n>1950049404+530$. Because $n-1$ has a representation as a
sum of $717$ positive ninth powers, then $n=\sum_{i=1}^{717}x_{i}^{9}+1^{9}$
is a representation of $n$ as the sum of $718$ positive ninth powers. It is
left to check for which values of $n<1950049404+530$ are not the sum of $718$
positive ninth powers:

$\mathbf{B}_{718}^{9}=\\{1,2,\ldots,717\\}\cup\\{718+\beta\mid\beta\in\mathbf{B}^{9}\\}.$

Note that
$\mathbf{B}_{718}^{9}=\\{\beta+1\mid\beta\in\mathbf{B}_{717}^{9}\\}\cup\\{1\\}$.
The largest value in $\mathbf{B}_{718}^{9}$ is $6465115$. Considering the
conditions in Theorem 4.18, $m=6465114,k=9$, and $j=717$, the first condition
is satisfied, and the second condition evaluates to

$\left\lfloor\sqrt[9]{6464397+\frac{6464397}{2^{8}-1}}\right\rfloor=\left\lfloor\sqrt[9]{6465114}\right\rfloor.$

∎

Because Vaughan & Wooley [78] showed that $G(9)\leq 50$ for sufficiently large
positive integers, the proof proceeds using the conjectured largest value that
cannot be expressed as the sum of $50$ positive ninth powers. If Conjecture
3.36 does not hold, then a substitution of the correct smallest such value is
easily performed without any loss of validity to the proof.

Note that $982835352$ works in the proof of Theorems 3.37, & 3.38 because it
is small enough for modest computers to find consecutive expressions as the
sum of $67,68,\ldots,187$ positive ninth powers, but there are (almost
certainly) larger numbers that would improve these results. How far can this
technique work for positive ninth powers?

There are almost certainly not positive integers that are the sum of
$2,3,4,\ldots,122$ positive ninth powers, considering a density argument, but
perhaps such a number exists yet is inaccessibly large. If so, this would
allow proof of which numbers are not the sum of $j\geq 123$ positive ninth
powers. This is the theoretical limit that this proof technique will allow for
positive ninth powers without further improvements in Waring’s problem. One
possible improvement that requires more investigation is the following.

######  Conjecture 11.1.

Every positive integer greater than $967214052$ is the sum of exactly $121$
positive ninth powers.

##  12\. Final Thoughts

For higher powers, where $k>9$, almost certainly something analogous occurs.
Eventually, $j$ and $n$ become sufficiently large that there are enough small
positive $k$th powers to piece together a representation of $n$. There is
every reason to believe that higher powers contain similar patterns, and there
appears to be much room for further investigation.

One line of inquiry that seems natural to consider is what happens when the
summands $x_{i}^{k}$ are restricted by some parameter.

######  Question 12.1.

What positive integers cannot be written as the sum of distinct positive $k$th
powers?

######  Question 12.2.

What positive integers cannot be written as the sum of prime positive $k$th
powers?

## Disclosure Statement

None of the authors have received any financial support regarding this
manuscript.

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