<s>[INST] Let a, b, c, x and y be integers. Why are there no solutions for $\sqrt {a^2 + b^2 + c^2} = x$, where $x=2^y$ or $x=5 \times 2^y$ Edit - An addition condition: $a$, $b$, and $c$ do not equal $0$.
I'm really digging into 3D vectors and their properties. I've decided to look and see which combinations of three integers will, when plotted in 3D space, generate a hypotenuse whose length is also an integer. For example; $\sqrt {6^2 + 42^2 + 85^2} = 95$
What I've found is quite interesting. It seems that there are no solutions for $\sqrt {a^2 + b^2 + c^2} = x$ when $x=2^y$ or $x=5 \times 2^y$ and $y$ is an integer.
$$\begin{array}{c|c|c} 
y & 2^y & 5 \times 2^y \\ \hline
0 & 1 & 5 \\ \hline
1 & 2 & 10 \\ \hline
2 & 4 & 20 \\ \hline
3 & 8 & 40 \\ \hline
4 & 16 & 80 \\ \hline
5 & 32 & 160 \\ \hline
6 & 64 & \text{see below} \\ \hline
7 & 128 & \text{see below} \\ 
\end{array}$$
So, there are no solutions for $\sqrt {a^2 + b^2 + c^2} = x$ when $x$ equals any value from the following series: $1,2,4,5,8,10,16,20,32,40,64,80,128,160...$
I wrote a program which brute-force checked this for values up to $250$ for $a, b, \text {and } c$ (which is why I show no solutions to $5 \times 2^x$ using $6$ and $7$ above, since they should be $320$ and $640$ respectively). The program found at least 1 solution for all other integers from $1 \text{ to }250$.
Here's the code I used (excel VBA): http://pastebin.com/jVA27jYp
 [/INST] WLOG, $a,b,c,y$ are positive. If there exists a solution to $a^2+b^2+c^2\in \{4^y, 25\cdot   4^y\}$ then  for some $y$ then there is a solution with a least value of  $\min (a,b,c).$ Such a solution cannot have $a,b,c$ all even, else $(a/2)^2+(b/2)^2+(c/2)^2\in \{4^{y-1},4^{y-1}\cdot 25\},$  which contradicts the minimality of $\max (a,b,c)$, unless $y=1$.But there is no solution with $y=1.$
But if $a,b,c$ are not all even then $a^2+b^2+c^2$ is congruent to $1,2,$ or $3$ (modulo $4$) so it cannot be divisible by $4,$ so it can't belong to $\{4^y,25\cdot 4^y\}.$ 
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