# rational ${\theta}$-parallelogram envelopes via ${\theta}$-congruent
elliptic curves

Sajad Salami and Arman Shamsi Zargar  Instítuto da Matemática e Estatística,
Universidade Estadual do Rio de Janeiro (UERJ), Rio de Janeiro, Brazil
<EMAIL_ADDRESS>Department of Mathematics and Applications,
University of Mohaghegh Ardabili, Ardabil, Iran<EMAIL_ADDRESS>
###### Abstract.

We introduce a new generalization of ${\theta}$-congruent numbers by defining
the notion of rational ${\theta}$-parallelogram envelope for a positive
integer $n$, where ${\theta}\in(0,\pi)$ is an angle with rational cosine.
Then, we study more closely some problems related to the rational
${\theta}$-parallelogram envelopes, using the arithmetic of algebraic curves.
Our results generalize the recent work of T. Ochiai, where only the case
${\theta}=\pi/2$ was considered. Moreover, we answer the open questions in his
paper and their generalizations for any Pythagorean angle.

Subjclass 2020: Primary 11G05; Secondary 14H52

keywords: Rational ${\theta}$-parallelogram envelope, ${\theta}$-congruent
number, elliptic curve.

##  1\. Introduction

A positive integer $n$ is called a congruent number if it is equal to the area
of a right triangle with rational sides. Equivalently, if there exist positive
rational numbers $a$, $b$, and $c$ such that $a<b<c,$ and

$a^{2}+b^{2}=c^{2},\quad ab=2n.$ (1.1)

Determining all the congruent numbers is an old problem in Number Theory.
There are various sorts of generalizations of the problem, see for example [3,
4, 10, 15, 16, 17].

In [3], M. Fujiwara introduced and studied an interesting generalization of
congruent numbers, called ${\theta}$-congruent numbers. Afterwards, several
authors ([1, 4, 6, 9, 11, 12]) studied this new concept with different
approaches using the arithmetic of elliptic curves [10, 14]. In order to
describe this generalization, let ${\theta}\in(0,\pi)$ be an angle with
rational cosine, i.e., it satisfies $\cos({\theta})=s/r$ with
$r,s\in{\mathbb{Z}}$ such that $0\leq|s|<r$ and $\gcd(r,s)=1$. Let
${\mathbb{N}}$ be the set of natural numbers. An element $n$ of ${\mathbb{N}}$
is called a ${\theta}$-congruent number if there exists a triangle with
rational sides and area equal to $n\sqrt{r^{2}-s^{2}}$. Equivalently, if there
are three positive rational numbers $a$, $b$, and $c$ satisfying $a\leq b<c$,
and

$a^{2}+b^{2}-\frac{2s}{r}ab=c^{2},\quad ab=2rn.$

We denote such a triangle by $(a,b,c)$ and call it a rational
${\theta}$-triangle for $n$. It is clear that if a positive integer $n$ is
${\theta}$-congruent with a ${\theta}$-triangle $(a,b,c)$, then $nm^{2}$ is
also a ${\theta}$-congruent number with the ${\theta}$-triangle $(ma,mb,mc)$.
Hence, one may concentrate on the square-free positive integers. The problem
of determining ${\theta}$-congruent numbers is related to finding
non-2-torsion points on the elliptic curve given by the following Weierstrass
equation,

$E_{\theta}^{n}:y^{2}=x(x+(r+s)n)(x-(r-s)n),$

where $r$ and $s$ are as above. One can see [3, 4, 9] for more details.

In a recent work, T. Ochiai [12] generalized the congruent numbers in a novel
way. Indeed, he considered the set ${\mathcal{N}}$ of all positive integers
$n$ such that there is an envelope for $n$, i.e., a quintuple $(a,b,c,d,e)$ of
positive rational numbers satisfying

$a^{2}+b^{2}=c^{2},\quad a^{2}+d^{2}=e^{2},\quad a(b+d)=n.$ (1.2)

We note that any congruent number belongs to the set ${\mathcal{N}}$, because
if we let $b=d$, then Equations (1.2) are equivalent to (1.1) for $4n$ and
therefore for $n$. It is proved that $n\in{\mathcal{N}}$ if either $n$ or $2n$
is a congruent number. Moreover, it is shown that $n\in{\mathcal{N}}$ if and
only if a certain set of algebraic equations has some rational solutions, see
Theorem 2 in [12]. Given any positive rational number $m$, T. Ochiai also
considered the set ${\mathcal{N}}(m)$ consisting of all positive integers
$n\in{\mathcal{N}}$ such that there exists an envelope $(a,b,c,d,e)$ for $n$
with $d=mb$. By studying the set of rational points on certain family of
elliptic curves and a genus five algebraic curve, he provided certain
conditions which lead to deciding whether a given positive integer $n$ belongs
to ${\mathcal{N}}(m)$. In the end of his paper [12], T. Ochiai asked the
following questions concerning the sets ${\mathcal{N}}$ and
${\mathcal{N}}(m)$.

######  Question 1.1.

Notation being as above, one may ask that:

  * (i)

Is ${\mathcal{N}}={\mathbb{N}}$?

  * (ii)

Given any $n\in{\mathcal{N}}$, are there infinitely many distinct envelopes
for $n$?

  * (iii)

Given any $n\in{\mathcal{N}}$, are there infinitely many positive rational
numbers $m$ such that $n\in{\mathcal{N}}(m)$?

In this paper, by generalizing the notion of envelope in T. Ochai's paper
[12], we introduce naturally a real-life geometric object, namely, the
rational ${\theta}$-parallelogram envelope for $n$. This new notion can be
viewed as a generalization of ${\theta}$-congruent numbers which involves the
rational ${\theta}$-triangles. We will study the set of
${\theta}$-parallelogram envelopes for positive numbers by investigating the
set of rational points on a certainly defined algebraic curve. Moreover, we
shall respond affirmatively to Question 1.1 in a more general setting related
to the set of ${\theta}$-parallelogram envelopes which arises through this
study.

##  2\. The rational ${\theta}$-parallelogram envelopes

Let ${\theta}\in(0,\pi)$ be an angle with rational cosine, i.e.,
$\cos({\theta})=s/r$ with $r,s\in{\mathbb{Z}}$ such that $0\leq|s|<r$ and
$\gcd(r,s)=1$. We call ${\theta}$ a Pythagorean angle if its sine is also
rational, say $\sin({\theta})=t/r$ for which $0\leq|t|=\sqrt{r^{2}-s^{2}}<r$
and $\gcd(r,t)=1$. Then, we start by the following definition.

######  Definition 2.1.

Let ${\theta}\in(0,\pi)$ be angle with rational cosine. We define
${\mathcal{N}}_{\theta}$ to be the set of all $n\in{\mathbb{N}}$ such that
there exist positive rational numbers $a,b,c,d,$ and $e$ satisfying

$a^{2}+b^{2}-\frac{2s}{r}ab=c^{2},\quad a^{2}+d^{2}+\frac{2s}{r}ad=e^{2},\quad
a(b+d)=rn.$ (2.1)

For any $n\in{\mathcal{N}}_{\theta}$, we denote such a quintuple of positive
rational numbers satisfying Equations (2.1) by $(a,b,c,d,e)_{\theta}$ and call
it a rational ${\theta}$-parallelogram envelope for $n$. This notion with
${\theta}=\pi/2$ can be viewed as a generalization of the envelopes defined by
T. Ochiai in [12]. Figure 1 shows a ${\theta}$-parallelogram envelope for $n$.

Figure 1. A rational ${\theta}$-parallelogram envelope for $n$

######  Remark 2.2.

The following facts can be easily deduced from Definition 2.1.

  * (1)

A natural number $n$ belongs to ${\mathcal{N}}_{\theta}$ if and only if
$nm^{2}\in{\mathcal{N}}_{\theta}$ for any integer $m$. Hence, the area of a
rational ${\theta}$-parallelogram envelope for $n$ is equal to
$n\sqrt{r^{2}-s^{2}}\bmod({\mathbb{Q}}^{*})^{2}$.

  * (2)

A quintuple $(a,b,c,d,e)_{\theta}$ is a rational ${\theta}$-parallelogram
envelope for $n$ if and only if the quintuple $(a,d,e,b,c)_{\theta}$ is a
rational $(\pi-{\theta})$-parallelogram envelope for $n$.

  * (3)

If $n$ is both ${\theta}$\- and $(\pi-{\theta})$-congruent number, then
$n\in{\mathcal{N}}_{\theta}$, but the converse is not true. For instant, $n=7$
is neither a $\pi/3$-congruent number nor a $2\pi/3$-congruent number, but
there is a rational $\pi/3$-parallelogram envelope for $n=7$ given by

$(a,b,c,d,e)_{\frac{\pi}{3}}=\left(\frac{143}{42},\frac{20}{7},\frac{19}{6},\frac{1256}{1001},\frac{3583}{858}\right)_{\frac{\pi}{3}}.$

By the above remarks, we restrict our study to the case ${\theta}\in(0,\pi/2]$
and the square-free natural numbers $n$ throughout this work. Furthermore, by
a ${\theta}$-parallelogram envelope for an $n$ we always mean the rational
one.

In the next definition, we consider a subset of ${\mathcal{N}}_{{\theta}}$
which involves the ${\theta}$-parallelogram envelope $(a,b,c,d,e)_{{\theta}}$
having an angle $\tau$ with rational cosine between the sides $c$ and $e$.

######  Definition 2.3.

Let $\tau\in(0,\pi)$ be an angle with rational cosine. We denote by
${\mathcal{N}}_{\theta}(\tau)$ the set of all $n\in{\mathcal{N}}_{\theta}$
such that there exist positive rational numbers $a,b,c,d,$ and $e$ satisfying
(2.1) and

$\cos\tau=\frac{a^{2}-bd-sa(b-d)/r}{ce}.$

For any $n\in{\mathcal{N}}_{\theta}(\tau)$, we denote such a quintuple by
$(a,b,c,d,e)_{\theta}^{\tau}$ and call it a ${\theta}$-parallelogram envelope
with an angle $\tau$ for $n$.

Figure 2 shows a ${\theta}$-parallelogram envelope with an angle $\tau$ for an
element $n$ of ${\mathcal{N}}_{\theta}(\tau)$.

Figure 2. A ${\theta}$-parallelogram envelope with angle $\tau$ for $n$

The following theorem, proved in Section 7, gives a relation between the set
${\mathcal{N}}_{\theta}(\tau)$ and the $\tau$-congruent numbers.

######  Theorem 2.4.

Let ${\theta}\in(0,\pi/2]$ and $\tau\in(0,\pi)$ be angles with rational
cosines satisfying $\sin({\theta})/\sin(\tau)\equiv n^{\prime}\ \bmod\
({\mathbb{Q}}^{*})^{2}$ for some positive integer $n^{\prime}$. Then,
$n\in{\mathbb{N}}$ belongs to ${\mathcal{N}}_{\theta}(\tau)$ if and only if
$2nn^{\prime}$ is a $\tau$-congruent number. In particular,
$n\in{\mathcal{N}}_{{\theta}}({\theta})$ if and only if $2n$ is a
${\theta}$-congruent number. Equivalently, $n$ is a ${\theta}$-congruent
number if and only if $2n\in{\mathcal{N}}_{{\theta}}({\theta})$.

##  3\. The ${\theta}$-congruent number elliptic curve and the set
${\mathcal{N}}_{{\theta}}$

In this section, we provide an algebraically necessary and sufficient
condition for $n\in{\mathbb{N}}$ to be an element of ${\mathcal{N}}_{\theta}$
using the ${\theta}$-congruent number elliptic curves.

######  Theorem 3.1.

A natural number $n$ belongs to ${\mathcal{N}}_{\theta}$ if and only if the
equations,

$\left\\{\begin{array}[]{l}E_{\theta}^{w}:y^{2}=x(x+(r+s)w)(x-(r-s)w),\vspace{.2cm}\\\
F_{\theta}^{N}:v^{2}=u(u-(r+s)N)(u+(r-s)N),\vspace{.2cm}\\\
C_{R}:xv=uy,\end{array}\right.$ (3.1)

have a rational solution $(u,v,w,x,y)$ where $N=2n-w$, $yv\not=0$ and $0<w\leq
n$.

Taking $w=n$ in Theorem 3.1 leads to the fact that if a natural number $n$ is
simultaneously ${\theta}$\- and ($\pi-{\theta}$)-congruent number, then
$n\in{\mathcal{N}}_{\theta}$. The existence of a rational solution
$(u,v,w,x,y)$ in Theorem 3.1 means that there exists a natural number $0<w\leq
n$ such that both the elliptic curves $E_{\theta}^{w}$ and $F_{\theta}^{N}$
have some non-$2$-torsion points $(x,y)$ and $(u,v)$ satisfying $x/y=u/v$,
where the letter ``R'' in $C_{R}$ refers to the word ``Ratio". We note that
the ratio $b/d$ of a ${\theta}$-parallelogram envelope $(a,b,c,d,e)_{\theta}$
for $n$ depends on the parameter $w$. For $0<w\leq n$, the point $P_{w}$ moves
from the point $A_{1}$ to the middle point $P_{n}$ of the side $A_{1}A_{2}$,
see Figure 3 below.

Figure 3. The role of $w$ in ${\theta}$-parallelogram envelope for $n$

In order to reduce the number of variables in (3.1), we let $v=uy/x$ in
$F_{\theta}^{N}$ to obtain the following equivalence for Theorem 3.1.

######  Theorem 3.2.

A natural number $n$ belongs to ${\mathcal{N}}_{\theta}$ if and only if the
following equations,

$\left\\{\begin{array}[]{l}E_{\theta}^{w}:y^{2}=x(x+(r+s)w)(x-(r-s)w),\vspace{.2cm}\\\
G_{\theta}^{N}:z^{2}=\left(x^{2}+2s(N+w)x-(r^{2}-s^{2})w^{2}\right)^{2}+4(r^{2}-s^{2})N^{2}x^{2},\end{array}\right.$
(3.2)

have a rational solution $(w,x,y,z)$ where $N=2n-w$, $y\not=0$, and $0<w\leq
n$.

The proofs of Theorems 3.1 and 3.2 are included in Section 8

##  4\. The ${\theta}$-parallelogram envelopes with ratio $m$

In this section, we are going to study ${\mathcal{N}}_{{\theta},m}$ a subset
of ${\mathcal{N}}_{\theta}$ associated to a given positive rational number
$m$.

######  Definition 4.1.

For a positive rational number $m$, let ${\mathcal{N}}_{{\theta},m}$ denote
the subset of ${\mathcal{N}}_{\theta}$ consisting of all natural numbers $n$
such that there exist positive rational numbers $a$, $b$, $c$, $d$, $e$
satisfying (2.1) and $d=mb$. We call such a quintuple a
${\theta}$-parallelogram envelope with ratio $m$ for $n$ and denote it by
$(a,b,c,d,e)_{{\theta}}^{m}$, see Figure 4.

Figure 4. A ${\theta}$-parallelogram envelope with ratio $m$ for $n$

For a positive integer $m$, a natural number $n$ belongs to
${\mathcal{N}}_{{\theta},m}$ if and only if $nm$ is an element of
${\mathcal{N}}_{\pi-{\theta},m}$. Indeed, if $(a,b,c,d,e)_{{\theta}}^{m}$ is a
${\theta}$-parallelogram envelope with ratio $m$ for $n$, then we obtain a
$(\pi-{\theta})$-parallelogram envelope with ratio $m$ as
$(d,a,e,ma,mc)_{\pi-{\theta}}^{m}$ for $mn$, by multiplying the first equation
of (2.1) by $m^{2}$ and the third by $m$, see Figure 5.

Figure 5. Two ${\theta}$-parallelogram envelopes with ratio $m$ for $n$ and
$mn$

The following theorem is a consequence of Theorems 3.1 and 3.2, which is
proved in Section 8.

######  Theorem 4.2.

A natural number $n$ belongs to ${\mathcal{N}}_{{\theta},m}$ for some positive
rational number $m$ if and only if the simultaneous equations,

$\left\\{\begin{array}[]{l}E_{\theta}^{(m,n)}:\displaystyle
y^{2}=x\left(x+\frac{2n(r+s)}{m+1}\right)\left(x-\frac{2n(r-s)}{m+1}\right),\vspace{.2cm}\\\
G_{\theta}^{(m,n)}:z^{2}=x^{4}+b_{1}x^{3}+b_{2}x^{2}+b_{3}x+b_{4},\end{array}\right.$
(4.1)

have a rational solution $(x,y,z)$ with $y\not=0$, where

$\begin{array}[]{ll}b_{1}=8ns,&b_{2}=\displaystyle\frac{8n^{2}(2m^{2}r^{2}+4s^{2}m+3s^{2}-r^{2})}{(m+1)^{2}},\vspace{.2cm}\\\
b_{3}=-\displaystyle\frac{32n^{3}s(r^{2}-s^{2})}{(m+1)^{2}},&b_{4}=\displaystyle\frac{16n^{4}(r^{2}-s^{2})^{2}}{(m+1)^{4}}.\end{array}$

##  5\. An elliptic curve related to ${\mathcal{N}}_{{\theta},m}$

In order to investigate the set of rational points satisfying Equations (4.1),
we prove the following two results on the quartic curve $G_{\theta}^{(m,n)}$
in Section 9.

######  Theorem 5.1.

The quartic curve $G_{\theta}^{(m,n)}$ in the statement of Theorem 4.2 can be
birationally transformed into the elliptic curve given by the following cubic
equation,

${\mathcal{G}}_{\theta}^{m}:\
{\mathcal{Y}}^{2}={\mathcal{X}}^{3}+(r^{2}m^{2}+2s^{2}m+r^{2}){\mathcal{X}}^{2}+m^{2}(r^{2}-s^{2})^{2}{\mathcal{X}}.$

Moreover, for ${\theta}\in(0,\pi/2]$ with rational cosine, the rank
$r_{\theta}(m)$ of the Mordell-Weil group of rational points on
${\mathcal{G}}_{\theta}^{m}$ is at least one for all but finitely many
$m\in{\mathbb{Q}}$ with an independent point given by

${\mathcal{P}}=(-(r^{2}-s^{2})m^{2},s(r^{2}-s^{2})m^{2}(m+1)).$

Given ${\theta}\in(0,\pi/2]$ with $\cos({\theta})=s/r\in{\mathbb{Q}}^{*}$, we
define the following quantities:

$\begin{array}[]{l}M_{0}=m(r^{2}-s^{2}),\vspace{.2cm}\\\
M_{1}=r(m+r)\left(r(m+1)+2\sqrt{M_{0}}\right),\vspace{.2cm}\\\
M_{2}=r(m+r)\left(r(m+1)-2\sqrt{M_{0}}\right).\end{array}$ (5.1)

We note that $M_{0}$, $M_{1}$, and $M_{2}$ are all positive numbers. Evidently
$M_{0},$ and $M_{1}>0$, since $0\leq|s|<r$ and $m>0$. To show $M_{2}>0$, it is
enough to observe

$rm+r-2\sqrt{M_{0}}>0\Longleftrightarrow
r^{2}(m+1)^{2}>4m(r^{2}-s^{2})\Longleftrightarrow r^{2}(m-1)^{2}+4ms^{2}>0$

which is the case.

In the next result, we determine all possibilities for
${\mathcal{G}_{\theta}^{m}}(\mathbb{Q})_{\text{tors}}$, i.e., the torsion
subgroup of $\mathcal{G}_{\theta}^{m}$.

######  Theorem 5.2.

Keeping the above notations, we have:

${\mathcal{G}_{\theta}^{m}}(\mathbb{Q})_{\rm{tors}}\cong\left\\{\begin{array}[]{ll}\displaystyle\frac{\mathbb{Z}}{8\mathbb{Z}}&\text{if~{}}\sqrt{M_{0}}~{}\text{and
either}~{}\sqrt{M_{1}}~{}\text{or}~{}\sqrt{M_{2}}\in{\mathbb{Q}}^{*}\vspace{.2cm}\\\
\displaystyle\frac{\mathbb{Z}}{2\mathbb{Z}}\times\frac{\mathbb{Z}}{8\mathbb{Z}}&\text{if~{}}\sqrt{M_{0}},\sqrt{M_{1}},\text{and}~{}\sqrt{M_{2}}\in{\mathbb{Q}}^{*}\vspace{.2cm}\\\
\displaystyle\frac{\mathbb{Z}}{4\mathbb{Z}}~{}\text{or}~{}\frac{\mathbb{Z}}{2\mathbb{Z}}\times\frac{\mathbb{Z}}{4\mathbb{Z}}&\text{otherwise}.\end{array}\right.$

The points $(M_{0},\pm r(m+1)M_{0})$ are of order $4$ and points with the
following ${\mathcal{X}}$-coordinates

$\begin{array}[]{l}{\mathcal{X}}_{1}=M_{0}+\left(r(m+1)+\sqrt{M_{1}}\right)\sqrt{M_{0}},\vspace{.2cm}\\\
{\mathcal{X}}_{2}=M_{0}+\left(r(m+1)-\sqrt{M_{1}}\right)\sqrt{M_{0}},\vspace{.2cm}\\\
{\mathcal{X}}_{3}=M_{0}-\left(r(m+1)+\sqrt{M_{2}}\right)\sqrt{M_{0}},\vspace{.2cm}\\\
{\mathcal{X}}_{4}=M_{0}-\left(r(m+1)-\sqrt{M_{2}}\right)\sqrt{M_{0}},\end{array}$

are of order $8$ in the torsion subgroup of ${\mathcal{G}_{\theta}^{m}}$.

Moreover, for ${\theta}\neq\pi/2$, the cases $\mathbb{Z}/4\mathbb{Z}$,
$\mathbb{Z}/8\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$
happen for infinitely many $m$, but the case
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$ occurs only for finitely
many $m$.

The following table shows examples for the torsion subgroups of
$\mathcal{G}_{\theta}^{m}(\mathbb{Q})$ respect to the conditions in Theorem
5.2. We refer the reader to see Table 2 in the Appendix for more data on the
rank and cardinal number of the torsion subgroup of $\mathcal{G}_{\theta}^{m}$
for some $0<m<1$.

The discriminant and $j$-invariant of the elliptic curve
${\mathcal{G}}_{\theta}^{m}$ are given respectively as follows:

$\displaystyle\Delta_{\theta}^{m}$
$\displaystyle=16r^{2}(r^{2}-s^{2})^{4}(m+1)^{2}m^{4}\left(r^{2}m^{2}+2(2s^{2}-r^{2})m+r^{2}\right),$
$\displaystyle j_{\theta}^{m}$
$\displaystyle=256\frac{\left(r^{4}m^{4}+4r^{2}s^{2}m^{3}+(s^{4}+6r^{2}s^{2}-r^{4})m^{2}+4r^{2}s^{2}m+r^{4}\right)^{3}}{(m+1)^{2}(r^{2}-s^{2})^{4}\left(r^{2}m^{2}+2(2s^{2}-r^{2})m+r^{2}\right)m^{4}r^{2}}.$

It is easy to check the following isomorphisms over rational numbers,

${\mathcal{G}}_{\theta}^{m}\cong{\mathcal{G}}_{\theta}^{1/m},\quad\text{and}\quad{\mathcal{G}}_{\theta}^{-m}\cong{\mathcal{G}}_{\theta}^{-1/m}.$

Thus, we may assume the elliptic curves ${\mathcal{G}}_{\theta}^{m}$ with
$m\geq 1$ in the rest of paper.

We end this section by remarking that ${\mathcal{G}}_{{\pi}/{2}}^{m}$ is
isomorphic to the elliptic curve

$E(m):U^{2}=X(X-m^{2})(X-(m-1)(m+1)),$

by changing the variables $\mathcal{X}=X-m^{2}$ and $\mathcal{Y}=U$, which is
given and studied in [12].

Table 1. Examples for the torsion subgroups of $\mathcal{G}_{\theta}^{m}(\mathbb{Q})$ in Theorem 5.2 $\\{r,s,m\\}$ | $\\{\sqrt{M_{0}},\sqrt{M_{1}},\sqrt{M_{2}}\\}$ | ${\mathcal{G}_{\theta}^{m}}(\mathbb{Q})_{\text{tors}}$
---|---|---
$\\{2,1,2\\}$ | $\\{\sqrt{6},2\sqrt{9+3\sqrt{6}},2\sqrt{9-3\sqrt{6}}\\}$ | $\mathbb{Z}/4\mathbb{Z}$
$\\{2,1,3\\}$ | $\\{3,4\sqrt{7},4\\}$ | $\mathbb{Z}/8\mathbb{Z}$
$\\{2,1,1\\}$ | $\\{\sqrt{3},2\sqrt{3}+2,2\sqrt{3}-2\\}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$
$\\{25,7,1\\}$ | $\\{24,70,10\\}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$

##  6\. The main results related to ${\mathcal{N}}_{{\theta},m}$ and
${\mathcal{N}}_{\theta}$

Given ${\theta}\in(0,\pi/2]$ with $\cos({\theta})=s/r\in{\mathbb{Q}}$ and a
rational number $m\geq 1$, we have the following results on
${\mathcal{N}}_{{\theta},m}$.

######  Theorem 6.1.

  * (i)

If $\sqrt{m}\sin({\theta})$ and $\sqrt{m+1}\in{\mathbb{Q}}^{*}$, then
$\sqrt{m(r^{2}-s^{2})}\ \bmod\ ({\mathbb{Q}}^{*})^{2}$ belongs to
${\mathcal{N}}_{{\theta},m}$, and some points of order $8$ in
${\mathcal{G}}_{{\theta}}^{m}({\mathbb{Q}})$ give some
${\theta}$-parallelogram envelope with ratio $m$.

  * (ii)

Otherwise, there exists a ${\theta}$-parallelogram envelope
$(a,b,c,d,e)_{\theta}^{m}$ for $n\in{\mathcal{N}}_{{\theta},m}$ if and only if
$r_{\theta}(m)\geq 1.$

Using the above theorem, we obtain the next result.

######  Theorem 6.2.

  * (i)

The set ${\mathcal{N}}_{{\theta},m}$ is empty if and only if
$\sqrt{m}\sin({\theta})$ or $\sqrt{m+1}$ is not a rational number and
$r_{\theta}(m)=0$.

  * (ii)

${\mathcal{N}}_{{\theta},m}=\left\\{\sqrt{m(r^{2}-s^{2})}\ \bmod\
({\mathbb{Q}}^{*})^{2}\right\\}$ if and only if $\sqrt{m}\sin({\theta})$ and
$\sqrt{m+1}$ belong to ${\mathbb{Q}}^{*}$ and $r_{\theta}(m)=0.$

  * (iii)

The set ${\mathcal{N}}_{{\theta},m}$ has infinitely many elements if and only
if $r_{\theta}(m)\geq 1.$

######  Theorem 6.3.

  * (i)

A natural number $n$ belongs to ${\mathcal{N}}_{{\theta},m}$ if and only if
the curve

${\mathcal{C}}_{\theta}^{(m,n)}:\left\\{\begin{array}[]{l}{\mathcal{Y}}^{2}={\mathcal{X}}^{3}+(r^{2}m^{2}+2s^{2}m+r^{2}){\mathcal{X}}^{2}+m^{2}(r^{2}-s^{2})^{2}{\mathcal{X}\vspace{.2cm}}\\\
{\mathcal{Z}}^{2}=\displaystyle\frac{2n(r^{2}-s^{2})}{m+1}\left({\mathcal{Y}}+s(m+1){\mathcal{X}}\right)\vspace{.1cm}\\\
\qquad\times\left({\mathcal{X}}^{2}-(r-s)(ms-r){\mathcal{X}}-(r-s){\mathcal{Y}}\right)\vspace{.2cm}\\\
\qquad\times\left({\mathcal{X}}^{2}+(r+s)(ms-r){\mathcal{X}}+(r+s){\mathcal{Y}}\right),\\\
\end{array}\right.$

has a rational point $({\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}})$ with
${\mathcal{Y}}\neq 0$.

  * (ii)

For any natural number $n$, there exist only finitely many
${\theta}$-parallelogram envelopes with ratio $m$ for $n$.

The case ${\theta}=\pi/2$ of the next result gives a positive answer for all
parts of Question 1.1; in other words, all questions in Section 7 of the T.
Ochai's paper.

######  Theorem 6.4.

Given a Pythagorean angle ${\theta}\in(0,\pi/2]$, the following hold:

  * (i)

${\mathcal{N}}_{\theta}={\mathbb{N}}$.

  * (ii)

Given any $n\in{\mathcal{N}}_{\theta}$, there are infinitely many distinct
${\theta}$-parallelogram envelopes for $n$.

  * (iii)

Given any $n\in{\mathcal{N}}_{\theta}$, there are infinitely many rational
numbers $m\geq 1$ such that $n\in{\mathcal{N}}_{{\theta},m}$.

We note that the proof of Theorem 6.4 works only for the Pythagorean angles.
Hence, the interested reader may consider the following question as a further
research on the subject.

######  Question 6.5.

Are the statements of Theorem 6.4 true for any arbitrary angle
${\theta}\in(0,\pi/2]$ with rational cosine?

##  7\. Proof of Theorem 2.4

Let $\cos(\tau)=p/q\in{\mathbb{Q}}$ with $p,q\in{\mathbb{Z}}$ satisfying
$0<p<q$ and $\gcd(p,q)=1$. By the assumption, we have
$\sqrt{r^{2}-s^{2}}=n^{\prime}v^{2}\sqrt{q^{2}-p^{2}}$, where $n^{\prime}$ is
a positive integer and $v$ is a rational number.

If $n\in{\mathcal{N}}_{\theta}(\tau)$, then
$4n\in{\mathcal{N}}_{\theta}(\tau)$, and there exists a
${\theta}$-parallelogram envelope $(a,b,c,d,e)_{\theta}^{\tau}$ with angle
$\tau$ for $4n$, i.e., there exists a quintuple $(a,b,c,d,e)_{\theta}^{\tau}$
of positive rational numbers satisfying

$\begin{array}[]{ll}\displaystyle
c^{2}=a^{2}+b^{2}-\frac{2sab}{r},&a(b+d)=4rn,\vspace{.2cm}\\\ \displaystyle
e^{2}=a^{2}+d^{2}+\frac{2sab}{r},&\displaystyle\cos(\tau)=\frac{a^{2}-bd-4sa(b-d)/r}{ce}.\end{array}$

These equations imply that

$c^{2}+e^{2}-\frac{2pce}{q}=(b+d)^{2}.$

Hence, we have a rational $\tau$-triangle $(c,e,b+d)$ with area

$2n\sqrt{r^{2}-s^{2}}=2nn^{\prime}v^{2}\sqrt{q^{2}-p^{2}}.$

Dividing all sides by $v$ leads to a $\tau$-triangle $(c/v,e/v,(b+d)/v)$ with
area $2nn^{\prime}\sqrt{q^{2}-p^{2}}$, see Figure 6. This means that
$2nn^{\prime}$ is a $\tau$-congruent number as desired.

Figure 6. A $\tau$-triangle for $2nn^{\prime}$ obtained from a
${\theta}$-parallelogram envelope for $n$

Conversely, since $2nn^{\prime}$ is a $\tau$-congruent number, there exists a
rational $\tau$-triangle $(a,b,c)$ for $2nn^{\prime}$. Then, as described in
the following, one may find a ${\theta}$-parallelogram envelope with angle
$\tau$ for $n$ given by

$\left(\frac{rab}{2qcn^{\prime}},\frac{a^{2}}{2c}-\frac{(pn^{\prime}-s)ab}{2qcn^{\prime}},\frac{a}{2},\frac{c^{2}-a^{2}}{2c}+\frac{(pn^{\prime}-s)ab}{2qcn^{\prime}},\frac{b}{2}\right)_{\theta}^{\tau}.$
(7.1)

We first note that the $\tau$-triangle $(a,b,c)$ for $2nn^{\prime}$ satisfies

$c^{2}=a^{2}+b^{2}-\frac{2pab}{q},\quad ab=4qnn^{\prime}.$ (7.2)

Hence, to find a ${\theta}$-parallelogram envelope with angle $\tau$ for $4n$,
we have to solve

$\displaystyle a^{2}$ $\displaystyle=x^{2}+y^{2}-\frac{2sxy}{r},$ (7.3)
$\displaystyle b^{2}$ $\displaystyle=x^{2}+(c-y)^{2}+\frac{2sx(c-y)}{r},$
(7.4) $\displaystyle xc$ $\displaystyle=4rn.$ (7.5)

Substituting (7.3) into (7.4) and using (7.2), we get $a^{2}-cy-sab/r+scx/r=0$
which gives

$y=\frac{a^{2}}{c}-\frac{pab}{qc}+\frac{sx}{r}.$ (7.6)

From (7.2) and (7.5), we have $x=rab/(n^{\prime}qc)$ which implies
$y=a^{2}/c-(pn^{\prime}-s)ab/(qcn^{\prime})$ by applying (7.6). Therefore, we
have found the ${\theta}$-parallelogram envelope
$(x,y,a,c-y,b)_{{\theta}}^{\tau}$ with angle $\tau$ for $4n$. Dividing all
components by $2$ leads to a ${\theta}$-parallelogram envelope for $n$ defined
by (7.1), see Figure 7.

Figure 7. A ${\theta}$-parallelogram envelope with angle $\tau$ for $n$
obtained from a rational $\tau$-triangle for $2nn^{\prime}$

Checking that it is really a ${\theta}$-parallelogram envelope with angle
$\tau$ for $n$ is left as an exercise to the reader.

For $\tau=\pi/2$, we have a ${\theta}$-parallelogram envelope with angle
$\pi/2$ given by

$\left(\frac{rab}{2cn^{\prime}},\frac{a^{2}}{2c}+\frac{sab}{2cn^{\prime}},\frac{a}{2},\frac{c^{2}-a^{2}}{2c}-\frac{sab}{2cn^{\prime}},\frac{b}{2}\right)_{\pi/2}.$

In the case $\tau={\theta}$, we obtain the following ${\theta}$-parallelogram
envelope with angle ${\theta}$ given by

$\left(\frac{ab}{2c},\frac{a^{2}}{2c},\frac{a}{2},\frac{c^{2}-a^{2}}{2c},\frac{b}{2}\right)_{\theta}$

using a rational ${\theta}$-triangle $(a,b,c)$. The last statement is trivial
by Remark 2.2 (1) and the definition of $\tau$-congruent numbers. Therefore,
the proof of Theorem 2.4 is completed.

##  8\. Proofs of Theorems 3.1, 3.2, and 4.2

The proof of Theorem 3.1 is a consequence of the following one-to-one
correspondence between ${\mathcal{R}}_{\theta}$, the set of all rational
solutions of (3.1), and ${\mathcal{P}}_{\theta}$ the set of all
${\theta}$-parallelogram envelopes for $n$. Indeed, for any
$(u,v,w,x,y)\in{\mathcal{R}}_{\theta}$, we have a ${\theta}$-parallelogram
envelope for $n$ as follows:

$\begin{split}&a=\displaystyle\left|\frac{y}{2x}\right|=\left|\frac{v}{2u}\right|,\quad
b=\left|\frac{rwx}{y}\right|,\quad
c=\left|\frac{x^{2}+(r^{2}-s^{2})w^{2}}{2y}\right|,\vspace{.2cm}\\\
&d=\displaystyle\left|\frac{r(2n-w)u}{v}\right|,\quad
e=\left|\frac{u^{2}+(r^{2}-s^{2})(2n-w)^{2}}{2v}\right|.\end{split}$ (8.1)

Conversely, any ${\theta}$-parallelogram envelope $(a,b,c,d,e)_{\theta}$ for
$n$ corresponds to a rational solution $(u,v,w,x,y)$ defined by

$\begin{array}[]{lll}x\displaystyle=2a\left(a+c-\frac{s}{r}b\right),&\displaystyle
y=4a^{2}\left(a+c-\frac{s}{r}b\right),&w=\displaystyle\frac{2ab}{r},\vspace{.2cm}\\\
u\displaystyle=2a\left(a+e+\frac{s}{r}d\right),&\displaystyle
v=4a^{2}\left(a+e+\frac{s}{r}d\right).&\end{array}$

In order to show Theorem 3.2, we first assume that
$n\in{\mathcal{N}}_{\theta}$. Then, by Theorem 3.1 there exists a rational
solution $(u,v,w,x,y)\in{\mathcal{R}}_{\theta}$ of Equations (3.1) satisfying
$yv\not=0$ and $0<w\leq n$. Substituting $uy/x$ for $v$ into the equation of
$F_{\theta}^{N}$ gives that

$\frac{u^{2}y^{2}}{x^{2}}=u(u+(r-s)N)(u-(r+s)N),$

where $N=2n-w$. Multiplying both sides of the latter equation by $x^{2}$ and
then dividing by $u$, we obtain

$x^{2}u^{2}-2x^{2}Nsu-(r^{2}-s^{2})N^{2}x^{2}-y^{2}u=0.$

Substituting $y^{2}$ with $x^{3}+2swx^{2}-(r^{2}-s^{2})w^{2}x$ into the above
equation and then dividing both sides of the resulting equation by $x$ give a
quadratic equation in terms of $u,$

$xu^{2}-(x^{2}+2xs(N+w)-(r^{2}-s^{2})w^{2})u-(r^{2}-s^{2})N^{2}x=0.$

Solving this equation, we obtain

$\displaystyle u$ $\displaystyle=\frac{x^{2}+2xs(N+w)-(r^{2}-s^{2})w^{2}}{2x}$
$\displaystyle\quad\pm\frac{\sqrt{(x^{2}+2xs(N+w)-(r^{2}-s^{2})w^{2})^{2}+4(r^{2}-s^{2})N^{2}x^{2}}}{2x}.$
(8.2)

Since $x$, $u$, and $w$ are all rational, there exists some $z\in{\mathbb{Q}}$
such that

$z^{2}=(x^{2}+2xs(N+w)-(r^{2}-s^{2})w^{2})^{2}+4(r^{2}-s^{2})N^{2}x^{2}.$
(8.3)

Thus, we obtain the desired rational solution $(w,x,y,z)$ of Equation (3.2).
Conversely, if we assume that $(w,x,y,z)$ with $y\not=0$ and $0<w\leq n$ is a
solution of Equation (3.2), then we obtain a rational solution $(u,v,w,x,y)$
of (3.1), where $u$ is given by Equation (8.2) and $v=yu/x\not=0$,
respectively. Therefore, the proof of Theorem 3.2 is completed.

By the condition $d=mb$, we obtain that

$a(b+d)=rn\Longleftrightarrow
ab(m+1)=rn\Longleftrightarrow\frac{2ab}{r}=\frac{2n}{m+1},$

and the aforementioned correspondence with $w=2n/(m+1)$ implies that a natural
number $n$ belongs to ${\mathcal{N}}_{{\theta},m}$ if and only if the
simultaneous Equations (3.1) with $N=2nm/(m+1)$ have a rational solution
$(u,v,x,y)$ with $yv\not=0$. Now, using Theorem 3.2 and some simple algebraic
simplifications on Equation (8.3) we obtain the desired Equation (4.1). This
completes the proof of Theorem 4.2.

##  9\. Proofs of Theorems 5.1 and 5.2

We fix the natural number $n$, the rational number $m\geq 1$ and an angle
${\theta}\in(0,\pi/2]$ with rational cosine. One can transform the quartic
$G_{\theta}^{(m,n)}$ into the cubic ${\mathcal{G}}_{\theta}^{m}$ by the
following birational transformations:

$\left\\{\begin{array}[]{l}x\displaystyle=\frac{-2n(r^{2}-s^{2})\left({\mathcal{Y}}+s(m+1){\mathcal{X}}\right)}{(m+1){\mathcal{X}}({\mathcal{X}}+r^{2}-s^{2})},\vspace{.2cm}\\\
z\displaystyle=\frac{4n^{2}(r^{2}-s^{2})}{(m+1)^{2}{\mathcal{X}}({\mathcal{X}}+r^{2}-s^{2})^{2}}\big{(}{\mathcal{X}}^{3}+(2d_{2}-(r^{2}-s^{2})){\mathcal{X}}^{2}\vspace{.1cm}\\\
\qquad+\left(3d_{1}+2s(m+1){\mathcal{Y}}\right){\mathcal{X}}+(r^{2}-s^{2})d_{1}\big{)},\end{array}\right.$
(9.1)

where $d_{1}=m^{2}(r^{2}-s^{2})^{2}$ and $d_{2}=(r^{2}m^{2}+2s^{2}m+r^{2})$
are coefficients of the elliptic curve ${\mathcal{G}}_{\theta}^{m}$, and

$\left\\{\begin{array}[]{l}{\mathcal{X}}=\displaystyle\frac{-(r^{2}-s^{2})\left((m+1)^{2}x^{2}+4sn(m+1)^{2}x+(m+1)^{2}z-4n^{2}(r^{2}-s^{2})\right)}{2(m+1)^{2}x^{2}},\vspace{.2cm}\\\
{\mathcal{Y}}=\displaystyle\frac{(r^{2}-s^{2})\left(s(m+1)^{4}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}\right)}{2(m+1)^{3}x^{3}},\end{array}\right.$

where

$\displaystyle c_{0}$
$\displaystyle=-2n(r^{2}-s^{2})\left((m+1)^{2}z-4n^{2}(r^{2}-s^{2})\right),$
$\displaystyle c_{1}$
$\displaystyle=s(m+1)^{2}\left((m+1)^{2}z-12n^{2}(r^{2}-s^{2})\right),$
$\displaystyle c_{2}$
$\displaystyle=2n(m+1)^{2}\left(2d_{2}-3(r^{2}-s^{2})\right).$

The points $(x,z)=\left(0,\pm 2n(r^{2}-s^{2})(m+1)\right)$ on the quartic
$G_{\theta}^{(m,n)}$ map to the point at infinity $\infty=[0:1:0]$ on
${\mathcal{G}}_{\theta}^{m}$ by the above change of variables. The Mordell-
Weil group of rational points on ${\mathcal{G}}_{\theta}^{m}$ has generic rank
one, because one can easily examine by SAGE software that $[n]{\mathcal{P}}$
are not $\infty$ for $2\leq n\leq 16$, where

${\mathcal{P}}=(-(r^{2}-s^{2})m^{2},s(r^{2}-s^{2})m^{2}(m+1)).$

Hence, the point ${\mathcal{P}}$ is of infinite order on
${\mathcal{G}}_{\theta}^{m}({\mathbb{Q}})$. This completes the proof of
Theorem 5.1.

In what follows we classify the possibilities for
${{\mathcal{G}}_{\theta}^{m}(\mathbb{Q})}_{\text{tors}}$. It has trivially the
point $(0,0)$, which is of order $2$. Hence, by the celebrated Mazur's theorem
[14] on the torsion subgroup of elliptic curves over the rational numbers, we
have the following possibilities:

$\frac{{\mathbb{Z}}}{2{\mathbb{Z}}},\ \frac{{\mathbb{Z}}}{4{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{6{\mathbb{Z}}},\ \frac{{\mathbb{Z}}}{8{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{10{\mathbb{Z}}},\ \frac{{\mathbb{Z}}}{12{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\times\frac{{\mathbb{Z}}}{4{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\times\frac{{\mathbb{Z}}}{6{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\times\frac{{\mathbb{Z}}}{8{\mathbb{Z}}}.$

For any point ${\mathcal{T}}=(\mathcal{X},\mathcal{Y})$ on the curve
$\mathcal{G}_{\theta}^{m}$, the duplication formula of
${\mathcal{G}}_{\theta}^{m}(\mathbb{Q})$ leads to

$\displaystyle{\mathcal{X}}([2]{\mathcal{T}})$
$\displaystyle=\frac{\left({\mathcal{X}}^{2}-(r^{2}-s^{2})^{2}m^{2}\right)^{2}}{{4\mathcal{Y}}^{2}}.$
(9.2)

Hence the $\mathcal{X}$-coordinates of the points $[2n]{\mathcal{T}}$, for
$n=2,3,4,5,6$, are also squares.

If ${\mathcal{T}}$ is a point of order $4$, then
${\mathcal{X}}([2]{\mathcal{T}})=0$. By the duplication formula and the fact
that $(0,0)$ is the unique point of order $2$ on ${\mathcal{G}}_{\theta}^{m}$,
we find out two points of order $4$ as ${\mathcal{T}}$ and $-{\mathcal{T}}$,
where

${\mathcal{T}}:=\left((r^{2}-s^{2})m,r(r^{2}-s^{2})m(m+1)\right).$

Thus, the cases $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/6\mathbb{Z}$ and
$\mathbb{Z}/10\mathbb{Z}$ cannot happen. Hence, the possibilities for torsion
subgroup reduce to

$\frac{{\mathbb{Z}}}{4{\mathbb{Z}}},\ \frac{{\mathbb{Z}}}{8{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{12{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\times\frac{{\mathbb{Z}}}{4{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\times\frac{{\mathbb{Z}}}{6{\mathbb{Z}}},\
\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\times\frac{{\mathbb{Z}}}{8{\mathbb{Z}}}.$

Let us rule out the two possibilities $\mathbb{Z}/12\mathbb{Z}$ and
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$. To this end, it suffices
to show the non-existence of any point of order $3$. By contrary, we assume
that ${\mathcal{T}}=({\mathcal{X}},{\mathcal{Y}})$ is such a point. Then, the
equality $[2]{\mathcal{T}}=-{\mathcal{T}}$ implies that

$-3{\mathcal{X}}^{4}+(-4r^{2}-8ms^{2}-4m^{2}r^{2}){\mathcal{X}}^{3}-6m^{2}(r-s)^{2}(r+s)^{2}{\mathcal{X}}^{2}+m^{4}(r-s)^{4}(r+s)^{4}=0,$

which is unsolvable over ${\mathbb{Q}}$. Hence, the cases
${\mathbb{Z}}/12{\mathbb{Z}}$ and
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/6{\mathbb{Z}}$ can never happen
as torsion subgroup of ${\mathcal{G}}_{\theta}^{m}$ over ${\mathbb{Q}}$.
Considering these observations and applying the Mazur's theorem show that

${\mathcal{G}}_{\theta}^{m}({\mathbb{Q}})_{\text{tors}}\cong\frac{{\mathbb{Z}}}{4{\mathbb{Z}}},~{}\text{or}~{}\frac{{\mathbb{Z}}}{8{\mathbb{Z}}},~{}\text{or}~{}\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\times\frac{{\mathbb{Z}}}{4{\mathbb{Z}}}~{}\text{or}~{}\frac{{\mathbb{Z}}}{2{\mathbb{Z}}}\times\frac{{\mathbb{Z}}}{8{\mathbb{Z}}}.$

Now, if we assume that ${\mathcal{T}}$ is a point of order $8$, then
$[2]{\mathcal{T}}$ (resp. $[4]{\mathcal{T}}$) will be a point of order $4$
(resp. $2$). Solving ${\mathcal{X}}([2]{\mathcal{T}})=(r^{2}-s^{2})m$ is
equivalent to finding the solutions of the following quartic equation,

$\displaystyle{\mathcal{X}}^{4}-4m(r^{2}-s^{2}){\mathcal{X}}^{3}-2m(r^{2}-s^{2})(2r^{2}+mr^{2}+2m^{2}r^{2}+3ms^{2}){\mathcal{X}}^{2}$
$\displaystyle\quad-4m^{3}(r^{2}-s^{2})^{3}{\mathcal{X}}+m^{4}(r^{2}-s^{2})^{4}=0,$

which can be rewritten as
$({\mathcal{X}}-m(r^{2}-s^{2}))^{4}=4mr^{2}(r^{2}-s^{2})(m+1)^{2}{\mathcal{X}}^{2}$.
Solving this equation leads to the following list of
${\mathcal{X}}$-coordinates of ${\mathcal{T}}$,

$\begin{array}[]{ll}{\mathcal{X}}_{1}=M_{0}+\left(r(m+1)+\sqrt{M_{1}}\right)\sqrt{M_{0}},\vspace{.2cm}\\\
{\mathcal{X}}_{2}=M_{0}+\left(r(m+1)-\sqrt{M_{1}}\right)\sqrt{M_{0}},\vspace{.2cm}\\\
{\mathcal{X}}_{3}=M_{0}-\left(r(m+1)+\sqrt{M_{2}}\right)\sqrt{M_{0}},\vspace{.2cm}\\\
{\mathcal{X}}_{4}=M_{0}-\left(r(m+1)-\sqrt{M_{2}}\right)\sqrt{M_{0}},\end{array}$

where $M_{0},M_{1}$, and $M_{2}$ are given by (5.1). Using the duplication
formula (9.2), one may write down the ${\mathcal{Y}}$-coordinates as follow:

$\begin{array}[]{l}{\mathcal{Y}}_{1}=\frac{1}{2}M_{0}\left(r(m+1)+\sqrt{M_{1}}\right)\left(r(m+1)+\sqrt{M_{1}}+2\right),\vspace{.2cm}\\\
{\mathcal{Y}}_{2}=\frac{1}{2}M_{0}\left(r(m+1)-\sqrt{M_{1}}\right)\left(r(m+1)-\sqrt{M_{1}}+2\right),\vspace{.2cm}\\\
{\mathcal{Y}}_{3}=\frac{1}{2}M_{0}\left(r(m+1)+\sqrt{M_{2}}\right)\left(r(m+1)+\sqrt{M_{2}}-2\right),\vspace{.2cm}\\\
{\mathcal{Y}}_{4}=\frac{1}{2}M_{0}\left(r(m+1)-\sqrt{M_{2}}\right)\left(r(m+1)+\sqrt{M_{2}}-2\right).\end{array}$

Thus, in order to have
${\mathcal{G}}_{\theta}^{m}({\mathbb{Q}})_{\text{tors}}\cong{\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/8{\mathbb{Z}}$,
we need to assume that all the numbers $\sqrt{M_{0}},\sqrt{M_{1}},$ and
$\sqrt{M_{2}}$ are rational. If one of the values $\sqrt{M_{1}}$ and
$\sqrt{M_{2}},$ does not belong to ${\mathbb{Q}}$, then
${\mathcal{G}}_{\theta}^{m}({\mathbb{Q}})_{\text{tors}}\cong{\mathbb{Z}}/8{\mathbb{Z}}$
provided that $\sqrt{M_{0}}\in{\mathbb{Q}}^{*}$. When the last condition fails
to be true, we will have
${\mathcal{G}}_{\theta}^{m}({\mathbb{Q}})_{\text{tors}}\cong{\mathbb{Z}}/4{\mathbb{Z}}\
\text{or}\ {\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}.$

In order to prove the last statement of Theorem 5.2, we assume that
$M_{0}=m_{0}^{2}$ is a rational square. Then $m=m_{0}^{2}/(r^{2}-s^{2}),$
which yields

$\displaystyle M_{1}$
$\displaystyle=\frac{r(m_{0}^{2}+r^{2}-s^{2})\left(r(m_{0}^{2}+(r^{2}-s^{2}))+2m_{0}(r^{2}-s^{2})\right)}{(r^{2}-s^{2})^{2}},$
$\displaystyle M_{2}$
$\displaystyle=\frac{r(m_{0}^{2}+r^{2}-s^{2})\left(r(m_{0}^{2}+(r^{2}-s^{2}))-2m_{0}(r^{2}-s^{2})\right)}{(r^{2}-s^{2})^{2}}.$

We let $m_{i}=m_{0}^{2}(r^{2}-s^{2})\sqrt{M_{i}}$ for $i=1,2.$ Then,

$\displaystyle m_{1}^{2}$
$\displaystyle=r(m_{0}^{2}+r^{2}-s^{2})\left(r(m_{0}^{2}+(r^{2}-s^{2}))+2m_{0}(r^{2}-s^{2})\right),$
(9.3) $\displaystyle m_{2}^{2}$
$\displaystyle=r(m_{0}^{2}+r^{2}-s^{2})\left(r(m_{0}^{2}+(r^{2}-s^{2}))-2m_{0}(r^{2}-s^{2})\right).$
(9.4)

Each of Equations (9.3) and (9.4) defines an elliptic curve over
${\mathbb{Q}}$, which is birational to

$E_{0}:Y^{2}=X^{3}-108\,{r}^{2}({r}^{2}-{s}^{2})^{2}({r}^{2}+3\,{s}^{2})X+432\,{r}^{4}(r^{2}-s^{2})^{3}(r^{2}-9s^{2}),$

by the following change of variables:

$\left\\{\begin{array}[]{l}X=\displaystyle\frac{6r\left(r(r^{2}-s^{2})m_{0}^{2}+3\varepsilon(r^{2}-s^{2})m_{0}+3(m_{i}+r)\right)}{m_{0}^{2}},\vspace{.2cm}\\\
Y=\displaystyle\frac{54r(r^{2}-s^{2})m_{0}\left(r(r^{2}-s^{2})^{2}m_{0}^{2}+2\varepsilon
r^{2}m_{0}+(m_{i}+3r)\right)+2\varepsilon
r(m_{i}+r)}{m_{0}^{3}},\end{array}\right.$

and

$\left\\{\begin{array}[]{l}m_{0}\displaystyle=\frac{6r\left(Y+3\varepsilon(r^{2}-s^{2})(X+12r^{2}(r^{2}-s^{2})\right)}{\left(X+12r^{2}(r^{2}-s^{2})\right)\left(X-24r^{2}(r^{2}-s^{2})\right)},\vspace{.2cm}\\\
m_{i}\displaystyle=\frac{r\left(X^{3}+e_{1}X+e_{0}\right)}{\left(X+12r^{2}(r^{2}-s^{2})\right)\left(X-24r^{2}(r^{2}-s^{2})\right)^{2}},\end{array}\right.$

where $\varepsilon=(-1)^{i-1}$ for $i=1,2,$ and

$\displaystyle e_{0}$
$\displaystyle=-72r^{2}(r^{2}-s^{2})^{2}\left(48r^{4}(r^{2}-s^{2})+\varepsilon
Y\right),$ $\displaystyle e_{1}$
$\displaystyle=108r^{2}(r^{2}-s^{2})(5r^{2}-9s^{2})\left((r^{2}-s^{2})+\varepsilon
Y\right).$

The torsion subgroup of $E_{0}$ is
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$ containing the
point at infinity and

$P_{0}=(-12r^{2}(r^{2}-s^{2}),0),\ P_{1}=(-6r(r-s)(r^{2}-9s^{2}),0),\
P_{2}=(-6r(r+3s)(r^{2}-s^{2}),0).$

Moreover, the Mordell-Weil rank of $E_{0}({\mathbb{Q}})$ is at least one with
the point of infinite order

$Q=\left(-3(r^{2}-s^{2})(r^{2}+3s^{2}),27(r^{2}-s^{2})^{3}\right).$

By the above change of variables, the point $(m_{0},m_{i})=(0,r(r^{2}-s^{2}))$
on Equations (9.3) and (9.4) gives the point at infinity on $E_{0}$, and the
points $P_{0}$ and $Q-P_{0}$ give two points at infinity on the quartic curves
(9.3) and (9.4). For a positive integer $\ell$, the $\ell$-multiplication
$[\ell]Q$ of $Q$ gives either the rational solution $(m_{0},m_{1})$ or
$(m_{0},m_{2})$. Hence, each of $\sqrt{M_{1}}$ and $\sqrt{M_{2}}$ can be
rational numbers infinitely many times, when the multiplications of $Q$ vary.
Moreover, the torsion points $P_{1}$ and $P_{2}$ are the only ones that give a
number $m_{0}$ leading to the rational numbers $m_{1}$, $m_{2}$ and hence the
rational numbers $\sqrt{M_{1}}$ and $\sqrt{M_{2}}$. This argument justifies
the last statement of Theorem 5.2.

##  10\. Proofs of the main results on ${\mathcal{N}}_{{\theta},m}$

###  10.1. Proof of Theorem 6.1

Let $\sqrt{m+1}=v$ and $\sqrt{m}\sin({\theta})=N^{\prime}u^{2}$, where
$N^{\prime}$ is a positive integer and $\alpha,u\in{\mathbb{Q}}$. Then,
$\sqrt{M_{0}}=N^{\prime}ru^{2}$ and

$\sqrt{M_{1}}=\frac{rv}{u}\sqrt{(v/u)^{2}+2N^{\prime}},\quad\sqrt{M_{2}}=\frac{rv}{u}\sqrt{(v/u)^{2}-2N^{\prime}}.$

Hence, both the numbers $\sqrt{M_{1}}$ and $\sqrt{M_{2}}$ are rational if and
only if there are rational numbers $\alpha$ and $\beta$ such that
$(v/u)^{2}+2N^{\prime}=\alpha^{2}$ and $(v/u)^{2}-2N^{\prime}=\beta^{2},$
which is equivalent to saying that $2N^{\prime}$ is a congruent number with a
right triangle $(\alpha-\beta,\alpha+\beta,2\gamma)$, where $\gamma=v/u$.
Hence, for such a right triangle we have $\sqrt{M_{1}}=r\alpha\gamma$ and
$\sqrt{M_{2}}=r\beta\gamma.$ In order to prove the part (i) of Theorem 6.1,
one needs to do cumbersome computations using a series of transformations
which we only give a sketch and leave the detailed computations to the reader.
For $i=1,\ldots,4$, writing the order $8$ points
$({\mathcal{X}}_{i},\pm{\mathcal{Y}}_{i})$ in terms of
$N^{\prime},u,v,\alpha,\beta$, transforming them into the points
$(x_{i},z_{i})$ on $G_{\theta}^{(m,n)}$ by (9.1), and finding
$y_{i}$-coordinates on $E_{\theta}^{(m,n)}$ using $x_{i}$'s, we obtain
rational solutions $(x_{i},y_{i},z_{i})$ of Equation (4.1). Calculating
$u_{i}$ from $x_{i}$ and $z_{i}$ using the formula (8.2) and then putting
$v_{i}=u_{i}y_{i}/x_{i}$ lead to the solutions $(u_{i},v_{i},w,x_{i},y_{i})$
with $w=2N^{\prime}/(N^{\prime}u^{2}+1)$ of Equations (3.2). Now, using the
correspondence (8.1), one may obtain a ${\theta}$-parallelograms envelope with
ratio $m$ for $\sqrt{M_{0}}\ \bmod\ ({\mathbb{Q}}^{*})^{2}$, which means that
it belongs to ${\mathcal{N}}_{{\theta},m}$ as desired in the part (i) of
Theorem 6.1.

If one or both $\sqrt{m+1}$ and $\sqrt{m}\sin({\theta})$ do not belong to
${\mathbb{Q}}$, then at least one of the $\sqrt{M_{i}}$ will be a non-rational
number, hence the torsion subgroup of ${\mathcal{G}}_{\theta}^{m}$ cannot have
a point of order $8$. Thus, it will be isomorphic to
${\mathbb{Z}}/4{\mathbb{Z}}$ or
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$, by Theorem 5.2.
The order $4$ points in ${\mathcal{G}}_{\theta}^{m}({\mathbb{Q}})_{tors}$ are
$\pm{\mathcal{T}}=((r^{2}-s^{2})m,\pm r(r^{2}-s^{2})m(m+1))$, which lead to a
rational point $(x,z)$ on $G_{\theta}^{(m,n)}$ with $x$-coordinate
$-2n(r+s)/(m+1)$. But, this is a zero of the cubic polynomial defining
$E_{\theta}^{(m,n)}$. Thus no points of order $4$ in
${\mathcal{G}}_{\theta}^{m}({\mathbb{Q}})_{\rm{tors}}$ give us a
${\theta}$-parallelogram envelope. Therefore, the existence of a
${\theta}$-parallelogram envelope with ration $m$ for $n$, which one or both
$\sqrt{m+1}$ and $\sqrt{m}\sin({\theta})$ are not rational, means that
${\mathcal{G}}_{\theta}^{m}({\mathbb{Q}})$ has a point of infinite order.

Conversely, we let $m$ and $\theta$ be as before, and $n$ be a natural number.
Given any point
$({\mathcal{X}},{\mathcal{Y}})\in{\mathcal{G}}_{\theta}^{(m,n)}({\mathbb{Q}})$
of infinite order, the ${\mathcal{Y}}$-coordinates cannot be either $0$ or
$\pm r(m+1){\mathcal{X}}$ since it is neither of order $2$ nor $4$. By
substituting

$x=\frac{-2n(r^{2}-s^{2})({\mathcal{Y}}+s(m+1){\mathcal{X}})}{(m+1){\mathcal{X}}({\mathcal{X}}+r^{2}-s^{2})}$

one can write the defining equation of $E_{\theta}^{(m,n)}$ as

$y^{2}=\left(\frac{2n(r^{2}-s^{2})}{(m+1){\mathcal{X}}({\mathcal{X}}+r^{2}-s^{2})}\right)^{2}\cdot
A_{\theta}^{(m,n)},$

where

$\displaystyle A_{\theta}^{(m,n)}$
$\displaystyle=\frac{2n\left({\mathcal{Y}}+s(m+1){\mathcal{X}}\right)\left({\mathcal{X}}^{2}-(r-s)(ms-r){\mathcal{X}}-(r-s){\mathcal{Y}}\right)}{(m+1){\mathcal{X}}({\mathcal{X}}+r^{2}-s^{2})}$
$\displaystyle\quad\times\left({\mathcal{X}}^{2}+(r+s)(ms-r){\mathcal{X}}+(r+s){\mathcal{Y}}\right).$

If it is necessary, one can choose the sign of ${\mathcal{Y}}$ in a way such
that $A_{\theta}^{(m,n)}$ is a positive rational number. To see this, we may
assume $0<s<r$. Since $({\mathcal{X}},{\mathcal{Y}})$ is not of finite order
and ${\mathcal{G}}_{\theta}^{m}$ passes through the origin of the coordinate
system, we have ${\mathcal{X}}>0$ implying that
${\mathcal{Y}}+s(m+1){\mathcal{X}}>0$. Hence, the expression
$A_{\theta}^{(m,n)}$ is positive if and only if the following two expressions
are of the same sign,

$A_{1}:={\mathcal{X}}^{2}-(r-s)(ms-r){\mathcal{X}}-(r-s){\mathcal{Y}},\quad
A_{2}:={\mathcal{X}}^{2}+(r+s)(ms-r){\mathcal{X}}+(r+s){\mathcal{Y}}.$

For ${\mathcal{Y}}>0$ we have two cases:

  * (i)

$A_{1},A_{2}>0$: The relation $A_{1}>0$ holds if and only if

${\mathcal{X}}>(r-s)(ms-r)\quad\text{and}\quad{\mathcal{Y}}<\frac{{\mathcal{X}}({\mathcal{X}}-(r-s)(ms-r))}{r-s},$

and $A_{2}>0$ if and only if

${\mathcal{X}}>(r+s)(r-ms)\quad\text{and}\quad{\mathcal{Y}}>0$

or

${\mathcal{X}}>(r+s)(r-ms)\quad\text{and}\quad{\mathcal{Y}}>-\frac{{\mathcal{X}}({\mathcal{X}}+(r+s)(ms-r))}{r+s}.$

  * (ii)

$A_{1},A_{2}<0$: The relation $A_{1}<0$ holds if and only if

${\mathcal{X}}<(r-s)(ms-r)\quad\text{and}\quad{\mathcal{Y}}>0$

or

${\mathcal{X}}>(r-s)(ms-r)\quad\text{and}\quad{\mathcal{Y}}>\frac{{\mathcal{X}}({\mathcal{X}}-(r-s)(ms-r))}{r-s},$

and $A_{2}<0$ if and only if

${\mathcal{X}}<(r+s)(r-ms)\quad\text{and}\quad{\mathcal{Y}}<-\frac{{\mathcal{X}}({\mathcal{X}}+(r+s)(ms-r))}{r+s}.$

For ${\mathcal{Y}}<0$ we have two cases:

  * (i)

$A_{1},A_{2}>0$: The relation $A_{1}>0$ holds if and only if

${\mathcal{X}}>(r-s)(ms-r)\quad\text{and}\quad{\mathcal{Y}}<0$

or

${\mathcal{X}}<(r-s)(ms-r)\quad\text{and}\quad{\mathcal{Y}}<\frac{{\mathcal{X}}({\mathcal{X}}-(r-s)(ms-r))}{r-s},$

and $A_{2}>0$ if and only if

${\mathcal{X}}>(r+s)(r-ms)\quad\text{and}\quad{\mathcal{Y}}>-\frac{{\mathcal{X}}({\mathcal{X}}+(r+s)(ms-r))}{r+s}.$

  * (ii)

$A_{1},A_{2}<0$. The relation $A_{1}<0$ holds if and only if

${\mathcal{X}}<(r-s)(ms-r)\quad\text{and}\quad{\mathcal{Y}}>\frac{{\mathcal{X}}({\mathcal{X}}-(r-s)(ms-r))}{r-s}$

and $A_{2}>0$ if and only if

${\mathcal{X}}<(r+s)(r-ms)\quad\text{and}\quad{\mathcal{Y}}<0$

or

${\mathcal{X}}>(r+s)(r-ms)\quad\text{and}\quad{\mathcal{Y}}<-\frac{{\mathcal{X}}({\mathcal{X}}+(r+s)(ms-r))}{r+s}.$

Therefore, by Theorem 3.2, a natural number $n\in{\mathbb{N}}$ belongs to
${\mathcal{N}}_{{\theta},m}$ provided that $A_{\theta}^{(m,n)}$ is a rational
square number.

###  10.2. Proof of Theorem 6.2

The parts (i) and (ii) are direct consequences of Theorem 6.1. The necessity
of (iii) is clear by sufficiencies of (i) and (ii).

For the sufficiency of (iii), we assume that $r_{\theta}(m)\geq 1$ and
$({\mathcal{X}},{\mathcal{Y}})$ is a point of infinite order. Let
$\varphi:{\mathcal{G}}_{\theta}^{(m,n)}({\mathbb{Q}})\rightarrow{\mathbb{Q}}$
be defined by $\varphi(\infty)=0$ and

$\displaystyle\varphi({\mathcal{X}},{\mathcal{Y}})$
$\displaystyle=A_{\theta}^{(m,n)}\cdot\left(\frac{m+1}{2n}\right){\mathcal{X}}({\mathcal{X}}+r^{2}-s^{2})$
$\displaystyle=({\mathcal{Y}}+s(m+1){\mathcal{X}})\left({\mathcal{X}}^{2}-(r-s)(ms-r){\mathcal{X}}-(r-s){\mathcal{Y}}\right)$
$\displaystyle\quad\times\left({\mathcal{X}}^{2}+(r+s)(ms-r){\mathcal{X}}+(r+s){\mathcal{Y}}\right).$

We also define the map
$\pi:{\mathbb{Q}}\rightarrow{\mathbb{Q}}^{*}/({\mathbb{Q}}^{*})^{2}\cup\\{0\\}$
by

$\pi(q)=\left\\{\begin{array}[]{ll}q\ \bmod\ {\mathbb{Q}}^{*}&\text{\rm if}\
q\neq 0\vspace{.2cm}\\\ 0&\text{\rm if}\ q=0.\end{array}\right.$

Then, the set ${\mathcal{N}}_{{\theta},m}$ has infinite element if and only if
the image of ${\mathcal{G}}_{\theta}^{(m,n)}({\mathbb{Q}})$ under the
composite map $\pi\circ\varphi$ is infinite by the proof of the part (ii) of
Theorem 6.1. We define $\Sigma$ to be the following set

$\left\\{{\mathbb{Q}}(\varphi(P)):\
P=({\mathcal{X}},{\mathcal{Y}})\in{\mathcal{G}}_{\theta}^{(m,n)}({\mathbb{Q}})\right\\}.$

Then, the image of ${\mathcal{G}}_{\theta}^{(m,n)}({\mathbb{Q}})$ by
$\pi\circ\varphi$ is infinite if and only if $\Sigma$ is infinite. Because, if
we assume that $\Sigma$ is finite, then by the pigeonhole principal there is a
field ${\mathbb{K}}\in\Sigma$ and an infinite subset
$H\subset{\mathcal{G}}_{\theta}^{(m,n)}({\mathbb{Q}})$ such that
${\mathbb{K}}={\mathbb{Q}}(\varphi(P))$ for all $P\in H$. This implies that
the algebraic surface ${\mathcal{S}}$ defined by the equation
${\mathcal{Z}}^{2}=\varphi({\mathcal{X}},{\mathcal{Y}})$ has infinitely many
${\mathbb{K}}$-rational points
$({\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}})=({\mathcal{X}}(P),{\mathcal{Y}}(P),\sqrt{\varphi(P)})$
for $P\in H$. Furthermore, $P\in{\mathcal{G}}_{\theta}^{(m,n)}({\mathbb{Q}})$
so that the following algebraic curve

${\mathcal{C}}_{\theta}^{m}:\left\\{\begin{array}[]{l}{\mathcal{Z}}^{2}=\varphi({\mathcal{X}},{\mathcal{Y}}),\vspace{.2cm}\\\
{\mathcal{Y}}^{2}={\mathcal{X}}^{3}+(r^{2}m^{2}+2s^{2}m+r^{2}){\mathcal{X}}^{2}+m^{2}(r^{2}-s^{2})^{2}{\mathcal{X}},\end{array}\right.$

has also infinitely many ${\mathbb{K}}$-rational points as above. But, this is
a contradiction with the Faltings' theorem [8] on the set of rational points
on algebraic curves with genus $\geq 2$, since we have the following result:

######  Lemma 10.1.

The genus of ${\mathcal{C}}_{\theta}^{m}$ is equal to $9$.

###### Proof.

Let $\omega$ be the canonical sheaf of ${\mathcal{C}}_{\theta}^{m}$. By
Exercise (II.8.4.e) in [7], it is isomorphic to the invertible sheaf
${\mathcal{O}}(4)$, and hence
$\deg(\omega)=4\cdot\deg({\mathcal{C}}_{\theta}^{m}).$ Using the classical
version of the Bezout's theorem, see Proposition 8.4 in [5] or Example 1 on
page 198 of [13], the degree of ${\mathcal{C}}_{\theta}^{m}$ over
${\mathbb{C}}$ is equal to $4$ and so $\deg(\omega)=16$. As a consequence of
the Riemann-Roch theorem, it is well known that the degree of canonical sheaf
of any algebraic curve of genus $g$ is equal to $2g-2$, see Example 1.3.3 in
Chapter IV of [7]. Therefore, the genus of ${\mathcal{C}}_{\theta}^{m}$ is
equal to $9$. ∎

###  10.3. Proof of Theorem 6.3

Putting
${\mathcal{Z}}=(m+1){\mathcal{X}}({\mathcal{X}}+r^{2}-s^{2})y/(2n(r^{2}-s^{2}))$
and using the variable changes given by (9.1), the simultaneous equations
defining $E_{\theta}^{(m,n)}$ and $G_{\theta}^{(m,n)}$ give the space curve
${\mathcal{C}}_{\theta}^{(m,n)}$ given in the statement of Theorem 6.3. On the
other hand, it is easy to check that there exists a solution $(x,y,z)$ of
Equation (4.1) if and only if ${\mathcal{C}}_{\theta}^{(m,n)}$ has a solution
$({\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}})$ with ${\mathcal{Y}}\neq 0$. This
shows the part (i) of Theorem 6.3. By a similar argument as given in the proof
of Lemma 10.1, one can show that ${\mathcal{C}}_{\theta}^{(m,n)}$ is an
algebraic curve of genus $9$. Hence, it contains only finitely many rational
solutions. Therefore, applying Theorem 4.2 gives us the part (ii) of Theorem
6.3.

###  10.4. Proof of Theorem 6.4

Since we have supposed that ${\theta}$ is a Pythagorean angle with
$\cos({\theta})=s/r$, its sine is rational number, i.e.,
$\sin({\theta})=\sqrt{r^{2}-s^{2}}/r\in{\mathbb{Q}}^{*}$. First, we prove the
part (i) by the following two steps.

Step 1: Given a natural number $n$, if $(c,e,f)$ is a rational triangle with
area $T=2n\sqrt{r^{2}-s^{2}}$, then there exists a ${\theta}$-parallelogram
envelope $(a/2,b/2,c/2,d/2,e/2)_{{\theta}}$ for $n$, where

$a=\frac{4rn}{f},\quad b=\frac{f^{2}+c^{2}-e^{2}}{2f}+\frac{as}{r},\quad
d=\frac{f^{2}+e^{2}-c^{2}}{2f}-\frac{as}{r}$ (10.1)

Without losing the generality, we may assume that $c<e<f$ holds for the sides
of the triangle. In order to see that the quintuple $(a,b,c,d,e)$ is
${\theta}$-parallelogram envelope for $4n$, it is enough to check that
Equations (2.1) hold. It is clear that $a+b=f$ and hence $a(b+d)=af=4rn$,
which is the last equality in (2.1). Substituting $b$ and $d$, as given above,
into the first and second equations of (2.1) shows that

$\displaystyle a^{2}+b^{2}-c^{2}-\frac{2sab}{r}$
$\displaystyle=a^{2}+d^{2}-e^{2}+\frac{2sad}{r}$
$\displaystyle=64n^{2}(r^{2}-s^{2})+(c+e+f)(c+e-f)(c-e-f)(c-e+f).$

Letting $p=(c+e+f)/2$ and using the Heron's formula, we have

$\displaystyle 2n\sqrt{r^{2}-s^{2}}$ $\displaystyle=\sqrt{p(p-c)(p-e)(p-f)}$
$\displaystyle=\frac{1}{4}\sqrt{(c+e+f)(c+e-f)(e+f-c)(c-e+f)},$

in other words,

$64n^{2}(r^{2}-s^{2})+(c+e+f)(c+e-f)(c-e-f)(c-e+f)=0.$

Hence, the quintuple $(a,b,c,d,e)$ is ${\theta}$-parallelogram envelope for
$4n$. Dividing by $2$, we obtain $(a/2,b/2,c/2,d/2,e/2)_{{\theta}}$ for $n$ as
desired.

Step 2: For any positive rational number $T$ there exists a rational triangle
with area $T$.

Indeed, this fact is proved by N. J. Fine in Theorem 2 of [2]. Let us consider
the following genus one quartic curve

$C_{T}:y^{2}=T^{2}x^{4}+T^{2}x^{3}-x-1.$

Then, for any rational point $P=(x,y)$ with nonzero coordinates, we have a
rational triangle with area $T$ given by

$(c,e,f)=\left(\frac{y}{x},\frac{T^{2}x^{2}+1}{y},\frac{T^{2}x^{4}+1}{xy}\right).$

Indeed, if we let $\tau$ be the angle between $c$ and $e$, then using the
equation of $C_{T}$ we get that

$\cos({\theta})=\frac{c^{2}+e^{2}-f^{2}}{2ce}={\frac{{T}^{2}{x}^{2}-1}{{T}^{2}{x}^{2}+1}},\quad\sin({\theta})={\frac{2xT}{{T}^{2}{x}^{2}+1}},$

and hence the area of triangle $(c,e,f)$ is equal to $ce\sin({\theta})/2=T,$
as desired.

Now, clearly we have ${\mathcal{N}}_{{\theta}}\subseteq{\mathbb{N}}$. On the
other hand, given any natural number $n$ and any Pythagorean angle
${\theta}\in(0,\pi/2]$ with $\cos({\theta})=s/r$ and
$\sin({\theta})\in{\mathbb{Q}}^{*}$, letting $T=2n\sqrt{r^{2}-s^{2}}$ in the
above arguments leads to a rational triangle $(c,e,f)$ and hence
${\theta}$-parallelogram envelope $(a/2,b/2,c/2,d/2,e/2)_{{\theta}}$ for $n$
by Step 1. Thus, the proof of part (i) is completed.

In order to prove the part (ii), we shall to investigate a little bit more on
the curve $C_{T}$. Clearly, it contains the rational point $P_{0}=(-1,0)$ and
hence one can easily check that it is birational to the elliptic curve,

$E_{T}:Y^{2}=X^{3}+3T^{2}X-T^{2}(T^{2}-1),$

by the following maps

$\left\\{\begin{array}[]{ll}x=\displaystyle\frac{X+1}{T^{2}-X},&y=\displaystyle\frac{(T^{2}+1)Y}{(T^{2}-X)^{2}}\vspace{.2cm}\\\
X=\displaystyle\frac{T^{2}x-1}{x+1},&Y=\displaystyle\frac{(T^{2}+1)y}{(x+1)^{2}}.\end{array}\right.$

We note that the point $P_{0}=(-1,0)$ on $C_{T}$ corresponds to the point at
infinity on $E_{T}$. It is easy to see that $Q_{0}=(T^{2},T(T^{2}+1))$ is a
point on $E_{T}$ corresponding to the point at infinity on $C_{T}$. Moreover,
one can easily check that the point

$Q_{1}=\displaystyle\left(\frac{T^{2}}{4},\frac{T(T^{2}-8)}{8}\right)$

is another point on $C_{T}$ corresponding to the non-torsion point

$P_{1}=\displaystyle\left(\frac{T^{2}+4}{3T^{2}},\frac{2(T^{2}+1)(T^{2}-8)}{9T^{3}}\right).$

The point $P_{1}$ gives us the following triangle

$(c,e,f)_{1}=\displaystyle\left(\frac{T(T^{2}+16)}{2(T^{2}-8)},\frac{T^{6}+96T^{4}+256}{6T(T^{2}-8)(T^{2}+4)},\frac{2(T^{2}+1)(T^{2}-8)}{3T(T^{2}+4)}\right).$

In order to have a rational triangle with positive sides satisfying the
condition $c<e<f$, we may change their rules of $c,e,$ and $f$ or replace
$-P_{1}$ with $P_{1}$ if necessary. Then, for any integer $\ell\geq 1$, we let
$P_{\ell}=[\ell]P_{1}$ be the $\ell$-th multiple of $P_{1}$ and denote its
corresponding rational triangle by $(c,e,f)_{\ell}$. Letting
$T=2n\sqrt{r^{2}-s^{2}}$ and using Equations (10.1) one can obtain infinity
many distinct ${\theta}$-parallelogram envelopes for each $n\geq 1$ and for
any Pythagorean angle ${\theta}\in(0,\pi/2]$ with $\cos({\theta})=s/r$.
Therefore, the part (ii) is proved.

The last part is a direct consequence of the part (ii) if we define $m=b/d$
for the infinitely many ${\theta}$-parallelogram envelopes corresponding to
the points $[\ell]P_{1}$ for $\ell\geq 1$.

######  Remark 10.2.

For ${\theta}=\pi/2$ and another angle ${\theta}$ with $\cos({\theta})=3/5$,
using the proof of Theorem 6.4, we provide some ${\theta}$-parallelogram
envelopes for square-free natural numbers $1\leq n\leq 50$ in Tables 3 and 4,
respectively. In those tables, the bold numbers are neither ${\theta}$\- and
nor $\pi-{\theta}$-congruent numbers.

######  Example 10.3.

For ${\theta}=\pi/3$, all the square-free natural numbers $1\leq n\leq 50$
which are neither $\pi/3$\- nor $2\pi/3$-congruent numbers but belong to
${\mathcal{N}}_{\pi/3}$ are $n=2,3,7,26,31,$ and $38.$ See Table 5 in the
Appendix for corresponding $\pi/3$-parallelogram envelopes for all square-free
natural numbers $1\leq n\leq 50$, which are found by an ad hoc searching for
the components.

## Acknowledgments

The first named author would likes to thank for the hospitality of Institute
of Advanced Studies in Basic Sciences (IASBS) during his sabbatical year as a
postdoctoral researcher supported by the Iranian National Elites Foundation.
The second named author also thanks for the hospitality and partial financial
support of IASBS.

## Appendix

Table 2. More data on the elliptic curve $\mathcal{G}_{\theta}^{m}$ | $(r,s)=(2,1)$ | $(r,s)=(3,1)$ | $(r,s)=(4,1)$
---|---|---|---
$m$ | $r_{\theta}(m)$ | ${\mathcal{G}}_{{\theta}^{m}}({\mathbb{Q}})_{\rm{tors}}$ | $r_{\theta}(m)$ | ${\mathcal{G}}_{{\theta}^{m}}({\mathbb{Q}})_{\rm{tors}}$ | $r_{\theta}(m)$ | ${\mathcal{G}}_{{\theta}^{m}}({\mathbb{Q}})_{\rm{tors}}$
${1}/{2}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 0 | ${\mathbb{Z}}/8{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${1}/{3}$ | 0 | ${\mathbb{Z}}/8{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${1}/{4}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${1}/{5}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${1}/{6}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${1}/{7}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${1}/{8}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${1}/{9}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${1}/{10}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${2}/{3}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${2}/{5}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${2}/{7}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${2}/{9}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${3}/{4}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$
${3}/{5}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 0 | ${\mathbb{Z}}/8{\mathbb{Z}}$
${3}/{7}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${3}/{8}$ | 1 | ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${3}/{10}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${4}/{5}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${4}/{7}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$
${4}/{9}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${5}/{6}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${5}/{7}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${5}/{8}$ | 1 | ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${5}/{9}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${6}/{7}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${7}/{8}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${7}/{9}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${7}/{10}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${8}/{9}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$
${9}/{10}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 1 | ${\mathbb{Z}}/4{\mathbb{Z}}$ | 2 | ${\mathbb{Z}}/4{\mathbb{Z}}$
Table 3. Some $\pi/2$-parallelogram envelopes for square-free $1\leq n\leq 50$ $n$ | $(a,b,c,d,e)_{\frac{\pi}{2}}$
---|---
${\bf 1}$ | $\left(\frac{2}{5},{\frac{7}{60}},{\frac{5}{12}},{\frac{143}{60}},{\frac{29}{12}}\right)$
${{\bf 2}}$ | $\left(\frac{1}{2},{\frac{4}{15}},{\frac{17}{30}},{\frac{56}{15}},{\frac{113}{30}}\right)$
${{\bf 3}}$ | $\left({\frac{14}{13}},{\frac{2233}{2340}},{\frac{259}{180}},{\frac{29999}{16380}},{\frac{2677}{1260}}\right)$
$5$ | $\left({\frac{46}{29}},{\frac{57017}{22620}},{\frac{2323}{780}},{\frac{328559}{520260}},{\frac{30629}{17940}}\right)$
$6$ | $\left({\frac{17}{10}},{\frac{5474}{1665}},{\frac{2465}{666}},{\frac{6842}{28305}},{\frac{19441}{11322}}\right)$
$7$ | $\left({\frac{14700}{9259}},{\frac{15470431}{19443900}},{\frac{175277}{98700}},{\frac{66913}{18518}},{\frac{371}{94}}\right)$
${\bf 10}$ | $\left({\frac{30300}{19649}},{\frac{83810401}{59536470}},{\frac{310001}{148470}},{\frac{99760}{19649}},{\frac{260}{49}}\right)$
${\bf 11}$ | $\left({\frac{88572}{57715}},{\frac{742637383}{464721180}},{\frac{2122949}{958188}},{\frac{642917}{115430}},{\frac{1375}{238}}\right)$
$13$ | $\left({\frac{172380}{113059}},{\frac{2949868471}{1499162340}},{\frac{5512277}{2214420}},{\frac{1483027}{226118}},{\frac{2249}{334}}\right)$
$14$ | $\left({\frac{115836}{76145}},{\frac{1353665377}{630023730}},{\frac{2112881}{802578}},{\frac{537152}{76145}},{\frac{700}{97}}\right)$
$15$ | $\left({\frac{305100}{200923}},{\frac{9512390479}{4086773820}},{\frac{12605629}{4535820}},{\frac{3034185}{401846}},{\frac{3435}{446}}\right)$
${\bf 17}$ | $\left({\frac{502860}{332059}},{\frac{26336096911}{9822305220}},{\frac{26142077}{8489460}},{\frac{5674583}{664118}},{\frac{4981}{574}}\right)$
${\bf 19}$ | $\left({\frac{784092}{518755}},{\frac{64879403383}{21407981340}},{\frac{50173589}{14815212}},{\frac{9897613}{1037510}},{\frac{6935}{718}}\right)$
$21$ | $\left({\frac{1169532}{774835}},{\frac{145721314183}{43152110820}},{\frac{90433669}{24448788}},{\frac{16327227}{1549670}},{\frac{9345}{878}}\right)$
$22$ | $\left({\frac{704220}{466817}},{\frac{53034857569}{14942812170}},{\frac{29750513}{7714410}},{\frac{5150992}{466817}},{\frac{2684}{241}}\right)$
$23$ | $\left({\frac{1682220}{1115659}},{\frac{303628332991}{81599299260}},{\frac{154752077}{38544780}},{\frac{25733297}{2231318}},{\frac{12259}{1054}}\right)$
${\bf 26}$ | $\left({\frac{1372956}{911585}},{\frac{203809927777}{48137157510}},{\frac{79970801}{17795622}},{\frac{11877008}{911585}},{\frac{4420}{337}}\right)$
$29$ | $\left({\frac{4248732}{2823235}},{\frac{1962339124183}{413626513380}},{\frac{611798069}{122920212}},{\frac{82020323}{5646470}},{\frac{24505}{1678}}\right)$
$30$ | $\left({\frac{2432700}{1616849}},{\frac{644258613601}{131110285410}},{\frac{187110001}{36409410}},{\frac{24293280}{1616849}},{\frac{6780}{449}}\right)$
$31$ | $\left({\frac{5546892}{3687355}},{\frac{3353898590383}{659785804860}},{\frac{909668189}{171595788}},{\frac{114486937}{7374710}},{\frac{29915}{1918}}\right)$
${\bf 33}$ | $\left({\frac{7122060}{4736059}},{\frac{5541717534511}{1022136253380}},{\frac{1319930077}{234596340}},{\frac{156505767}{9472118}},{\frac{36069}{2174}}\right)$
$34$ | $\left({\frac{4012476}{2668625}},{\frac{1760698855777}{314935110750}},{\frac{394219121}{68094078}},{\frac{45425632}{2668625}},{\frac{9860}{577}}\right)$
${\bf 35}$ | $\left({\frac{9011100}{5993923}},{\frac{8888050331479}{1543195415580}},{\frac{1874280629}{314873580}},{\frac{210044765}{11987846}},{\frac{43015}{2446}}\right)$
$37$ | $\left({\frac{11253180}{7487059}},{\frac{13883190600871}{2277114124260}},{\frac{2610706277}{415759380}},{\frac{277325323}{14974118}},{\frac{50801}{2734}}\right)$
$38$ | $\left({\frac{6259740}{4165217}},{\frac{4298876938849}{686136196410}},{\frac{765244913}{118770330}},{\frac{79221488}{4165217}},{\frac{13756}{721}}\right)$
$39$ | $\left({\frac{13889772}{9243115}},{\frac{21179374244383}{3291916921020}},{\frac{3574266349}{540988812}},{\frac{360837633}{18486230}},{\frac{59475}{3038}}\right)$
$41$ | $\left({\frac{16964652}{11291275}},{\frac{31631011985983}{4672013439300}},{\frac{4817922509}{694723188}},{\frac{463356047}{22582550}},{\frac{69085}{3358}}\right)$
${\bf 42}$ | $\left({\frac{9340380}{6217217}},{\frac{9593473907809}{1382646888630}},{\frac{1390928113}{195925590}},{\frac{130672752}{6217217}},{\frac{18564}{881}}\right)$
${\bf 43}$ | $\left({\frac{20523900}{13662259}},{\frac{46341994593031}{6520996220700}},{\frac{6403414277}{881573100}},{\frac{587954437}{27324518}},{\frac{79679}{3694}}\right)$
$46$ | $\left({\frac{13438716}{8947505}},{\frac{19893681780577}{2613977795730}},{\frac{2395438961}{308798322}},{\frac{205938688}{8947505}},{\frac{24380}{1057}}\right)$
$47$ | $\left({\frac{29291340}{19503259}},{\frac{94544614798351}{12154821073980}},{\frac{10896327677}{1375446540}},{\frac{917276393}{39006518}},{\frac{104011}{4414}}\right)$
Table 4. Some ${\theta}$-parallelogram envelopes for square-free $1\leq n\leq 50$, where $\cos({\theta})=3/5$ $n$ | $[r_{1},t_{1}]$ | $[r_{2},t_{2}]$ |  $(a,b,c,d,e)_{{\theta}}$ with $\cos({\theta})=3/5$
---|---|---|---
$1$ | $[0,4]$ | $[1,4]$ | $\left({\frac{204}{91}},{\frac{25085}{18564}},{\frac{2561}{1428}},{\frac{80}{91}},{\frac{20}{7}}\right)$
${\bf 2}$ | $[0,4]$ | $[0,4]$ | $\left({\frac{15600}{7967}},{\frac{27143809}{12428520}},{\frac{90113}{48360}},{\frac{23288}{7967}},{\frac{136}{31}}\right)$
$3$ | $[1,4]$ | $[0,4]$ | $\left({\frac{78300}{40967}},{\frac{626820049}{213847740}},{\frac{870913}{370620}},{\frac{201432}{40967}},{\frac{444}{71}}\right)$
$5$ | $[0,4]$ | $[1,4]$ | $\left({\frac{601500}{318599}},{\frac{33244215601}{7665491940}},{\frac{16960001}{4787940}},{\frac{2837120}{318599}},{\frac{2020}{199}}\right)$
$6$ | $[1,4]$ | $[0,4]$ | $\left({\frac{249264}{132307}},{\frac{3943947275}{785223144}},{\frac{49766401}{11923128}},{\frac{206040}{18901}},{\frac{3480}{287}}\right)$
$7$ | $[1,4]$ | $[0,4]$ | $\left({\frac{2307900}{1226567}},{\frac{461284727569}{80879827980}},{\frac{124160513}{25782540}},{\frac{15820168}{1226567}},{\frac{5516}{391}}\right)$
$10$ | $[1,4]$ | $[0,4]$ | $\left({\frac{9606000}{5114399}},{\frac{7594358198401}{982578335880}},{\frac{1039360001}{153503880}},{\frac{96620440}{5114399}},{\frac{16040}{799}}\right)$
$11$ | $[0,4]$ | $[1,4]$ | $\left({\frac{2812524}{1497883}},{\frac{3217536636125}{382984716972}},{\frac{1836567041}{247246428}},{\frac{31291480}{1497883}},{\frac{21340}{967}}\right)$
$13$ | $[1,4]$ | $[0,4]$ | $\left({\frac{27428700}{14613767}},{\frac{8583302528647}{880959628380}},{\frac{4986522113}{570094980}},{\frac{51958816}{2087681}},{\frac{35204}{1351}}\right)$
${\bf 14}$ | $[0,4]$ | $[0,4]$ | $\left({\frac{7378224}{3931603}},{\frac{107881530976129}{10360088433240}},{\frac{7769251841}{825834072}},{\frac{528553256}{19658015}},{\frac{43960}{1567}}\right)$
$15$ | $[0,4]$ | $[1,4]$ | $\left({\frac{48613500}{25907399}},{\frac{186109696832401}{16792657883820}},{\frac{11741760001}{1166075820}},{\frac{748377960}{25907399}},{\frac{54060}{1799}}\right)$
$17$ | $[1,4]$ | $[0,4]$ | $\left({\frac{80197500}{42746567}},{\frac{500956576117489}{40331385964500}},{\frac{24845158913}{2180428500}},{\frac{1405736528}{42746567}},{\frac{78676}{2311}}\right)$
$19$ | $[0,4]$ | $[1,4]$ | $\left({\frac{25025964}{13340827}},{\frac{1208792909726929}{87859751640060}},{\frac{48375155201}{3802629372}},{\frac{2460337816}{66704135}},{\frac{109820}{2887}}\right)$
$21$ | $[2,4]$ | $[1,4]$ | $\left({\frac{37345644}{19909915}},{\frac{106891722534361}{7081415215812}},{\frac{88123230721}{6272289828}},{\frac{162797376}{3981983}},{\frac{148260}{3527}}\right)$
$22$ | $[0,4]$ | $[1,4]$ | $\left({\frac{224914800}{119911967}},{\frac{3864726317451649}{245181600685560}},{\frac{116460838913}{7914956280}},{\frac{5142200008}{119911967}},{\frac{170456}{3871}}\right)$
$23$ | $[1,4]$ | $[0,4]$ | $\left({\frac{268679100}{143248967}},{\frac{5498971329569809}{334678291560780}},{\frac{152018586113}{9885054540}},{\frac{6429413192}{143248967}},{\frac{194764}{4231}}\right)$
$26$ | $[1,4]$ | $[0,4]$ | $\left({\frac{87747504}{46786771}},{\frac{2910709595735885}{157900860594984}},{\frac{317031669761}{18248105928}},{\frac{2380601080}{46786771}},{\frac{281320}{5407}}\right)$
$29$ | $[0,4]$ | $[1,4]$ | $\left({\frac{135808044}{72416155}},{\frac{34652540443428529}{1695637304232900}},{\frac{610185464321}{31502783172}},{\frac{20595602336}{362080775}},{\frac{390340}{6727}}\right)$
${\bf 30}$ | $[0,4]$ | $[0,4]$ | $\left({\frac{777654000}{414669599}},{\frac{45368631579081601}{2149796482271640}},{\frac{747740160001}{37322207640}},{\frac{24416175720}{414669599}},{\frac{432120}{7199}}\right)$
$31$ | $[2,4]$ | $[1,4]$ | $\left({\frac{177327564}{94557787}},{\frac{11775813401066525}{540893613740028}},{\frac{910222297601}{43971515628}},{\frac{5756758280}{94557787}},{\frac{476780}{7687}}\right)$
$33$ | $[1,4]$ | $[0,4]$ | $\left({\frac{1138549500}{607130567}},{\frac{96798236787579889}{4189383051470100}},{\frac{1324284774913}{60108513300}},{\frac{39390937872}{607130567}},{\frac{575124}{8711}}\right)$
$34$ | $[1,4]$ | $[1,4]$ | $\left({\frac{256590384}{136827859}},{\frac{17534322811061047}{737578001737560}},{\frac{1583932334081}{69785037672}},{\frac{6536496328}{97734185}},{\frac{629000}{9247}}\right)$
$35$ | $[0,4]$ | $[1,4]$ | $\left({\frac{1440673500}{768251399}},{\frac{154572690545960401}{6324568182155580}},{\frac{1884688960001}{80669483580}},{\frac{52917310040}{768251399}},{\frac{686140}{9799}}\right)$
$37$ | $[1,4]$ | $[2,4]$ | $\left({\frac{1799276700}{959493767}},{\frac{240522746409818929}{9331863668963940}},{\frac{2630182554113}{106507454820}},{\frac{69927835648}{959493767}},{\frac{810596}{10951}}\right)$
$38$ | $[0,4]$ | $[1,4]$ | $\left({\frac{2001817200}{1067508767}},{\frac{297391044346297729}{11247144268059960}},{\frac{3086401626113}{121699949880}},{\frac{79934843912}{1067508767}},{\frac{878104}{11551}}\right)$
$39$ | $[0,4]$ | $[1,4]$ | $\left({\frac{444198924}{236879323}},{\frac{365692973639290129}{13489941076211340}},{\frac{3606747056641}{138578674572}},{\frac{91056034056}{1184396615}},{\frac{949260}{12167}}\right)$
$41$ | $[2,4]$ | $[1,4]$ | $\left({\frac{542566284}{289339099}},{\frac{15557821814308955}{546988291847868}},{\frac{4868447111681}{177948507828}},{\frac{3343084240}{41334157}},{\frac{1102900}{13447}}\right)$
$42$ | $[0,4]$ | $[1,4]$ | $\left({\frac{2987334000}{1593089567}},{\frac{659694548714076289}{22662336326401800}},{\frac{5625548070913}{200734619400}},{\frac{132034235928}{1593089567}},{\frac{1185576}{14111}}\right)$
$43$ | $[1,4]$ | $[0,4]$ | $\left({\frac{3282159900}{1750322567}},{\frac{795640144838322769}{26720179262662620}},{\frac{6478327040513}{225797335260}},{\frac{148566077752}{1750322567}},{\frac{1272284}{14791}}\right)$
${\bf 46}$ | $[0,4]$ | $[0,4]$ | $\left({\frac{859696944}{458467795}},{\frac{54456420261173161}{1713666792538776}},{\frac{9708557393921}{316349786328}},{\frac{8333017576}{91693559}},{\frac{1557560}{16927}}\right)$
$47$ | $[1,4]$ | $[0,4]$ | $\left({\frac{4684626300}{2498272967}},{\frac{1615810660236674449}{49802022322498860}},{\frac{11045411686913}{352263963180}},{\frac{232036850408}{2498272967}},{\frac{1661356}{17671}}\right)$
Table 5. Some $\pi/3$-parallelogram envelopes for square-free $1\leq n\leq 50$ $n$ | $[r_{1},t_{1}]$ | $[r_{2},t_{2}]$ | $(a,b,c,d,e)_{\pi/3}$
---|---|---|---
$1$ | $[0,8]$ | $[0,4]$ | $\left(\frac{5}{11},\frac{3}{8},\frac{37}{88},\frac{161}{40},\frac{1879}{440}\right)$
$\bf{2}$ | $[0,4]$ | $[0,4]$ | $\left(\frac{360}{209},\frac{567}{418},\frac{657}{418},\frac{9083}{9405},\frac{22183}{9405}\right)$
$\bf{3}$ | $[0,4]$ | $[0,4]$ | $\left(2,\frac{5}{4},\frac{7}{4},\frac{7}{4},\frac{13}{4}\right)$
$5$ | $[0,4]$ | $[1,4]$ | $\left(\frac{24}{7},\frac{5}{2},\frac{43}{14},\frac{5}{12},\frac{307}{84}\right)$
$6$ | $[1,4]$ | $[0,4]$ | $\left(\frac{8}{5},\frac{7}{6},\frac{43}{30},\frac{19}{3},\frac{109}{15}\right)$
${\bf{7}}$ | $[0,4]$ | $[0,4]$ | $\left(\frac{143}{42},\frac{20}{7},\frac{19}{6},\frac{1256}{1001},\frac{3583}{858}\right)$
$10$ | $[1,4]$ | $[1,4]$ | $\left(\frac{40}{7},\frac{15}{7},5,\frac{19}{14},\frac{13}{2}\right)$
$11$ | $[1,4]$ | $[0,4]$ | $\left(\frac{44}{15},\frac{11}{10},\frac{77}{30},\frac{32}{5},\frac{124}{15}\right)$
$13$ | $[1,4]$ | $[0,4]$ | $\left(\frac{13}{3},\frac{13}{8},\frac{91}{24},\frac{35}{8},\frac{181}{24}\right)$
$14$ | $[0,4]$ | $[2,4]$ | $\left(\frac{14}{5},\frac{7}{4},\frac{49}{20},\frac{33}{4},\frac{199}{20}\right)$
$15$ | $[0,4]$ | $[1,4]$ | $\left(6,\frac{9}{4},\frac{21}{4},\frac{11}{4},\frac{31}{4}\right)$
$17$ | $[1,4]$ | $[1,4]$ | $\left(\frac{15}{4},\frac{52}{15},\frac{217}{60},\frac{28}{5},\frac{163}{20}\right)$
$19$ | $[0,4]$ | $[1,4]$ | $\left(\frac{57}{40},\frac{9}{7},\frac{381}{280},\frac{533}{21},\frac{21943}{840}\right)$
$21$ | $[1,4]$ | $[1,4]$ | $\left(\frac{21}{4},\frac{14}{5},\frac{91}{20},\frac{26}{5},\frac{181}{20}\right)$
$22$ | $[1,4]$ | $[1,4]$ | $\left(8,3,7,\frac{5}{2},\frac{19}{2}\right)$
$23$ | $[1,4]$ | $[1,4]$ | $\left(\frac{92}{35},\frac{529}{220},\frac{3887}{1540},\frac{3321}{220},\frac{25513}{1540}\right)$
$\bf{26}$ | $[0,4]$ | $[0,4]$ | $\left(\frac{104}{15},\frac{104}{15},\frac{104}{15},\frac{17}{30},\frac{217}{30}\right)$
$29$ | $[0,4]$ | $[1,4]$ | $\left(\frac{15}{4},\frac{52}{15},\frac{217}{60},12,\frac{57}{4}\right)$
$30$ | $[1,4]$ | $[0,4]$ | $\left(\frac{60}{7},\frac{75}{14},\frac{15}{2},\frac{23}{14},\frac{19}{2}\right)$
$\bf{31}$ | $[0,4]$ | $[0,4]$ | $\left(\frac{1860}{343},\frac{10013}{2058},\frac{31}{6},\frac{11264}{1715},\frac{52}{5}\right)$
$33$ | $[0,4]$ | $[1,4]$ | $\left(\frac{60}{7},\frac{128}{17},\frac{964}{119},\frac{29}{170},\frac{10303}{1190}\right)$
$34$ | $[1,4]$ | $[1,4]$ | $\left(\frac{34}{15},\frac{34}{15},\frac{34}{15},\frac{416}{15},\frac{434}{15}\right)$
$35$ | $[1,4]$ | $[0,4]$ | $\left(\frac{140}{13},\frac{224}{39},\frac{28}{3},\frac{59}{78},\frac{67}{6}\right)$
$37$ | $[1,4]$ | $[0,4]$ | $\left(\frac{222}{35},\frac{333}{140},\frac{111}{20},\frac{3901}{420},\frac{817}{60}\right)$
$\bf{38}$ | $[0,4]$ | $[0,4]$ | $\left(\frac{3}{2},\frac{4}{5},\frac{13}{10},\frac{748}{15},\frac{1519}{30}\right)$
$39$ | $[2,4]$ | $[1,4]$ | $\left(\frac{65}{6},\frac{325}{48},\frac{455}{48},\frac{103}{240},\frac{2653}{240}\right)$
$41$ | $[1,4]$ | $[1,4]$ | $\left(\frac{60}{7},\frac{832}{105},\frac{124}{15},\frac{23}{14},\frac{19}{2}\right)$
$42$ | $[1,4]$ | $[0,4]$ | $\left(\frac{28}{5},\frac{7}{2},\frac{49}{10},\frac{23}{2},\frac{151}{10}\right)$
$43$ | $[0,4]$ | $[1,4]$ | $\left(\frac{14}{3},\frac{77}{20},\frac{259}{60},\frac{2041}{140},\frac{7303}{420}\right)$
$46$ | $[1,4]$ | $[0,4]$ | $\left(\frac{15}{2},\frac{104}{15},\frac{217}{30},\frac{16}{3},\frac{67}{6}\right)$
$47$ | $[1,4]$ | $[1,4]$ | $\left(\frac{423}{56},\frac{423}{56},\frac{423}{56},\frac{2465}{504},\frac{5473}{504}\right)$

In Tables 3–5, the highlighted $n$'s in boldface address they are neither
${\theta}$\- nor $(\pi-{\theta})$-congruent numbers. And in Tables 4 and 5,
the values $r_{1}$ and $r_{2}$ denote, resp., the rank of the (corresponding)
${\theta}$\- and $(\pi-{\theta})$-congruent elliptic curves over
${\mathbb{Q}}$, while $t_{1}$ and $t_{2}$, resp., refer to the number of
torsion points on the ${\theta}$\- and $(\pi-{\theta})$-congruent elliptic
curves.

## References

  * [1] A. Dujella, A. S. Janfada, C. J. Peral, S. Salami, On the high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves, Rocky Mountain J. Math. 44 (2014) 1867–1880.
  * [2] N. J. Fine, On rational triangles, Amer. Math. Month. 83 (1976) 517–521.
  * [3] M. Fujiwara, ${\theta}$-congruent numbers, in: Number Theory, K. Győry, A. Pethő and V. Sós (eds.), de Gruyter (1997) 235–241.
  * [4] M. Fujiwara, Some properties of ${\theta}$-congruent numbers, Natur. Sci. Rep. Ochanomizu Univ. 52 (2002) 1–8.
  * [5] W. Fulton, Intersection Theory, Second edition. Springer-Verlag, Berlin (1998).
  * [6] A. S. Janfada, S. Salami, On ${\theta}$-congruent numbers on real quadratic number fields, Kodai Math. J. 38 (2015) 352–364.
  * [7] R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag, New York (1977).
  * [8] H. Hindry, J. H. Silverman, Diophantine Geometry: An Introduction, GTM, 201, Springer-Verlag, New York (2001).
  * [9] M. Kan, ${\theta}$-congruent numbers and elliptic curves, Acta Arith. 94 (2000) 153–160.
  * [10] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, GTM 97, 2nd ed, Springer-Verlag, Berlin (1993).
  * [11] Y. Moranki, Adaptation of Monsky matrices for ${\theta}$-congruent numbers, Int. J. Number Theory 16 (2020) 154–170.
  * [12] T. Ochiai, New generalizations of congruent numbers, J. Number Theory, 193 (2018) 154–170.
  * [13] I. R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, Berlin-New York (1977).
  * [14] J. H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag, Berlin (2009).
  * [15] J. Top, N. Yui, Congruent number problems and their variants, in: Algorithmic number theory, Math. Sci. Res. Inst. Publ. 44 (2008) 613–639.
  * [16] S.-I. Yoshida, Some variant of the congruent number problem, I, Kyushu J. Math. 55 (2001) 387–404.
  * [17] S.-I. Yoshida, Some variant of the congruent number problem, II, Kyushu J. Math. 56 (2002) 147–165.