I'd like to ask whether there exists a formula which generates the primitive integer solutions to the Diophantine equation $x^2+y^2=2(u^2+v^2)$? Primitive means $x$, $y$, $u$ and $v$ are coprime. Thanks.
2026-03-26 13:01:23.1774530083
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Formula to generate primitive quadruples to Diophantine equation $x^2+y^2=2(u^2+v^2)$
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Choose any pair of of relative primes for $u$ and $v$, with one odd and the other even. Let $x=u+v$ and $y=u-v$. Based on $(a^2 +b^2)(c^2+d^2)=(ac+bd)^2 +(ad-bc)^2$ and $2=1^2+1^2$ I'm not suggesting this is exhaustive but it's simple.
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The given equation is equivalent to $\left| \frac{x+iy}{u+iv} \right| = \sqrt{2}$ (except for the trivial solution $x=y=u=v=0$). Therefore, if we can enumerate the rational points $z \in \mathbb{Q}[i]$ such that $|z| = \sqrt{2}$, we will be done. This is equivalent to finding the points $(a,b) \in \mathbb{Q}^2$ such that $a^2 + b^2 = 2$.
From here, we can use the standard procedure: for any such point $(a,b)$, the line joining $(a,b)$ to $(1,1)$ will have rational slope (or slope $\infty$); and conversely, any line through $(1,1)$ with rational slope will cut the circle in exactly one other point which will have rational coordinates (assuming you consider the tangent with slope $-1$ to cut $(1,1)$ twice). Now, to solve this: we have $b = m(a-1) + 1 = ma - m + 1$ so $a^2 + b^2 = a^2 + m^2 a^2 - 2(m-1) a + (m-1)^2 = (1+m^2) a^2 - 2m(m-1) a + (m-1)^2 = 2$. Now since we know $a=1$ is one root of this equation, the other root must be $\frac{2m(m-1)}{1+m^2} - 1 = \frac{m^2-2m-1}{m^2+1}$; and then $b = m(a-1) + 1 = \frac{-m^2-2m+1}{1+m^2}$. Now, if $m = \frac{p}{q}$, then $a = \frac{p^2-2pq-q^2}{p^2+q^2}$ and $b = \frac{-p^2-2pq+q^2}{p^2+q^2}$.
This gives rational solutions of the form $x + iy = (a+ib) (u+iv) = (au-bv) + (av+bu) i$, i.e. $x = au - bv$ and $y = bu + av$. Substituting in, and scaling up by the denominator, we get integer solutions of the form $x = (p^2-2pq-q^2)u_0 - (-p^2-2pq+q^2)v_0$, $y = (-p^2-2pq+q^2)u_0 + (p^2-2pq-q^2)v_0$, $u = (p^2+q^2) u_0$, $v = (p^2+q^2) v_0$. For any particular integer values of $p,q,u_0,v_0$, you can then calculate $x,y,u,v$ and then divide the solution $(x,y,u,v)$ by $\gcd(x,y,u,v)$ to get a primitive solution. This should then generate each primitive solution (and should even generate each primitive solution exactly once with the restriction that $\gcd(p,q)=\gcd(u,v)=1$).
Come to think of it, it should also be possible to start with $(u,v)$ on a circle of radius $r$, then $(u-v,u+v)$ is on the circle of radius $r\sqrt{2}$, and then find the other intersection of a line with slope $m$ with the circle of radius $r\sqrt{2}$. If you then proceed similarly, you might get a somewhat simpler expression for the primitive solutions.
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HINT.-A parametric solution of $$x^2+y^2=2z^2$$ is given by $$\begin{cases}x=a^2-b^2+2ab\\y=a^2-b^2-2ab\\z=a^2+b^2\end{cases}$$ Besides the pythagorean triple gives $$(a^2+b^2)^2=(a^2-b^2)^2+(2ab)^2$$ Finally we have the identity $$(a^2-b^2+2ab)^2+(a^2-b^2-2ab)^2=2[(a^2-b^2)^2+(2ab)^2]$$ from which we get infinitely many solutions of the proposed equation with the parameters $a$ and $b$.
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$$ \color{magenta}{ x = 2pr+qs+2qr \; , \; \; y = 2pr + qs + 2ps \; , \; \; u=2pr +qr +ps \; , \; \; v = qs +qr+ps \; \; .} $$
|x| |y| |u| |v| p q r s
1 1 1 0 0 -1 -1 1
2 0 1 1 0 -1 -1 0
3 1 2 1 0 -1 -2 1
4 2 3 1 0 -1 -3 2
5 1 3 2 0 -1 -3 1
5 3 4 1 0 -1 -4 3
5 5 4 3 1 -3 -1 3
6 4 5 1 0 -1 -5 4
7 1 4 3 0 -1 -4 1
7 1 5 0 1 1 -2 1
7 3 5 2 0 -1 -5 3
7 5 6 1 0 -1 -6 5
8 2 5 3 0 -1 -5 2
8 6 5 5 1 -4 -2 1
8 6 7 1 0 -1 -7 6
9 1 5 4 0 -1 -5 1
9 5 7 2 0 -1 -7 5
9 7 7 4 1 1 -2 -1
9 7 8 1 0 -1 -8 7
10 0 7 1 1 1 -3 2
10 4 7 3 0 -1 -7 4
10 8 9 1 0 -1 -9 8
11 1 6 5 0 -1 -6 1
11 3 7 4 0 -1 -7 3
11 3 8 1 1 1 -3 1
11 5 8 3 0 -1 -8 5
11 7 7 6 1 -5 -2 1
11 7 9 2 0 -1 -9 7
11 9 10 1 0 -1 -10 9
12 10 11 1 0 -1 -11 10
12 2 7 5 0 -1 -7 2
13 11 12 1 0 -1 -12 11
13 11 9 8 1 -3 -2 7
13 13 12 5 2 -5 -2 5
13 1 7 6 0 -1 -7 1
13 1 9 2 1 1 -4 3
13 3 8 5 0 -1 -8 3
13 5 9 4 0 -1 -9 5
13 7 10 3 0 -1 -10 7
13 9 10 5 1 1 -3 -1
13 9 11 2 0 -1 -11 9
14 12 11 7 1 -5 -3 2
14 12 13 1 0 -1 -13 12
14 4 9 5 0 -1 -9 4
14 8 11 3 0 -1 -11 8
14 8 9 7 1 -6 -2 1
15 11 13 2 0 -1 -13 11
15 13 14 1 0 -1 -14 13
15 1 8 7 0 -1 -8 1
15 5 11 2 1 1 -4 1
15 7 11 4 0 -1 -11 7
16 10 13 3 0 -1 -13 10
16 14 15 1 0 -1 -15 14
16 2 11 3 1 1 -5 4
16 2 9 7 0 -1 -9 2
16 6 11 5 0 -1 -11 6
17 11 13 6 1 1 -4 -1
17 11 14 3 0 -1 -14 11
17 1 12 1 1 1 -5 3
17 13 15 2 0 -1 -15 13
17 15 16 1 0 -1 -16 15
17 17 15 8 1 -5 -4 3
17 1 9 8 0 -1 -9 1
17 3 10 7 0 -1 -10 3
17 5 11 6 0 -1 -11 5
17 7 12 5 0 -1 -12 7
17 7 13 0 2 1 -3 1
17 9 11 8 1 -7 -2 1
17 9 13 4 0 -1 -13 9
18 16 13 11 1 -4 -5 3
18 16 17 1 0 -1 -17 16
18 4 11 7 0 -1 -11 4
18 4 13 1 1 1 -5 2
18 8 13 5 0 -1 -13 8
19 1 10 9 0 -1 -10 1
19 11 15 4 0 -1 -15 11
19 13 12 11 1 1 -7 9
19 13 16 3 0 -1 -16 13
19 15 17 2 0 -1 -17 15
19 17 15 10 1 1 -4 -3
19 17 17 6 2 1 -3 -1
19 17 18 1 0 -1 -18 17
19 3 11 8 0 -1 -11 3
19 3 13 4 1 1 -6 5
19 5 12 7 0 -1 -12 5
19 7 13 6 0 -1 -13 7
19 7 14 3 1 1 -5 1
19 9 11 10 1 3 -2 -1
19 9 14 5 0 -1 -14 9
20 10 13 9 1 -8 -2 1
20 14 17 3 0 -1 -17 14
20 18 19 1 0 -1 -19 18
20 2 11 9 0 -1 -11 2
20 6 13 7 0 -1 -13 6
21 1 11 10 0 -1 -11 1
21 11 16 5 0 -1 -16 11
21 1 14 5 1 3 -3 1
21 13 16 7 1 1 -5 -1
21 13 17 4 0 -1 -17 13
21 17 14 13 1 -3 -3 11
21 17 19 2 0 -1 -19 17
21 19 20 1 0 -1 -20 19
21 5 13 8 0 -1 -13 5
22 12 17 5 0 -1 -17 12
22 16 17 9 1 -7 -3 2
22 16 19 3 0 -1 -19 16
22 20 19 9 1 -6 -4 3
22 20 21 1 0 -1 -21 20
22 4 13 9 0 -1 -13 4
22 4 15 5 1 1 -7 6
22 8 15 7 0 -1 -15 8
23 11 15 10 1 -9 -2 1
23 11 17 6 0 -1 -17 11
23 11 18 1 2 1 -4 1
23 1 12 11 0 -1 -12 1
23 1 16 3 1 1 -7 5
23 13 18 5 0 -1 -18 13
23 15 16 11 2 -5 -2 7
23 15 19 4 0 -1 -19 15
23 17 20 3 0 -1 -20 17
23 19 18 11 1 1 -5 -3
23 19 21 2 0 -1 -21 19
23 21 17 14 1 -3 -4 13
23 21 22 1 0 -1 -22 21
23 3 13 10 0 -1 -13 3
23 5 14 9 0 -1 -14 5
23 7 15 8 0 -1 -15 7
23 7 17 0 1 3 -4 3
23 9 16 7 0 -1 -16 9
23 9 17 4 1 1 -6 1
24 10 13 13 1 -6 -3 1
24 10 17 7 0 -1 -17 10
24 14 19 5 0 -1 -19 14
24 2 13 11 0 -1 -13 2
24 2 17 1 1 1 -7 4
24 22 19 13 1 1 -5 -4
24 22 23 1 0 -1 -23 22
25 11 18 7 0 -1 -18 11
25 1 13 12 0 -1 -13 1
25 13 19 6 0 -1 -19 13
25 15 16 13 1 1 -9 11
25 15 19 8 1 1 -6 -1
25 17 21 4 0 -1 -21 17
25 19 18 13 1 -5 -5 3
25 19 22 3 0 -1 -22 19
25 21 22 7 2 1 -4 -1
25 21 23 2 0 -1 -23 21
25 23 24 1 0 -1 -24 23
25 25 24 7 3 -7 -3 7
25 3 14 11 0 -1 -14 3
25 5 17 6 1 1 -8 7
25 5 18 1 1 1 -7 3
25 7 16 9 0 -1 -16 7
25 9 17 8 0 -1 -17 9
26 0 17 7 1 4 -3 1
26 12 17 11 1 -10 -2 1
26 12 19 7 0 -1 -19 12
26 16 21 5 0 -1 -21 16
26 20 23 3 0 -1 -23 20
26 24 25 1 0 -1 -25 24
26 4 15 11 0 -1 -15 4
26 8 17 9 0 -1 -17 8
26 8 19 3 1 1 -7 2
27 11 16 13 1 3 -3 -1
27 11 19 8 0 -1 -19 11
27 11 20 5 1 1 -7 1
27 1 14 13 0 -1 -14 1
27 1 19 2 1 1 -8 5
27 13 20 7 0 -1 -20 13
27 17 22 5 0 -1 -22 17
27 19 17 16 1 1 -10 13
27 19 23 4 0 -1 -23 19
27 23 23 10 1 -7 -4 3
27 23 25 2 0 -1 -25 23
27 25 26 1 0 -1 -26 25
27 5 16 11 0 -1 -16 5
27 5 19 4 2 1 -5 3
27 7 17 10 0 -1 -17 7
28 10 19 9 0 -1 -19 10
28 10 21 1 1 3 -5 4
28 18 23 5 0 -1 -23 18
28 2 15 13 0 -1 -15 2
28 22 25 3 0 -1 -25 22
28 26 21 17 1 -4 -8 5
28 26 27 1 0 -1 -27 26
28 6 17 11 0 -1 -17 6
28 6 19 7 1 1 -9 8
29 11 16 15 1 -7 -3 1
29 11 20 9 0 -1 -20 11
29 1 15 14 0 -1 -15 1
29 13 19 12 1 -11 -2 1
29 13 21 8 0 -1 -21 13
29 15 22 7 0 -1 -22 15
29 15 23 2 2 1 -5 1
29 17 22 9 1 1 -7 -1
29 17 23 6 0 -1 -23 17
29 19 24 5 0 -1 -24 19
29 21 25 4 0 -1 -25 21
29 23 19 18 1 -3 -4 15
29 23 26 3 0 -1 -26 23
29 25 27 2 0 -1 -27 25
29 27 23 16 1 1 -6 -5
29 27 28 1 0 -1 -28 27
29 29 21 20 2 -7 -5 3
29 3 16 13 0 -1 -16 3
29 3 19 8 1 3 -4 1
29 3 20 5 1 1 -9 7
29 5 17 12 0 -1 -17 5
29 7 18 11 0 -1 -18 7
29 7 21 2 1 1 -8 3
29 9 19 10 0 -1 -19 9
30 16 17 17 1 -5 -5 2
30 16 23 7 0 -1 -23 16
30 20 19 17 1 1 -11 14
30 20 23 11 1 -9 -3 2
30 28 29 1 0 -1 -29 28
30 4 17 13 0 -1 -17 4
30 8 19 11 0 -1 -19 8
31 11 21 10 0 -1 -21 11
31 1 16 15 0 -1 -16 1
31 1 20 9 1 5 -3 1
31 13 22 9 0 -1 -22 13
31 13 23 6 1 1 -8 1
31 15 23 8 0 -1 -23 15
31 17 20 15 1 1 -11 13
31 17 24 7 0 -1 -24 17
31 17 25 0 3 1 -4 1
31 19 25 6 0 -1 -25 19
31 21 26 5 0 -1 -26 21
31 23 24 13 1 1 -7 -3
31 23 27 4 0 -1 -27 23
31 25 27 8 2 1 -5 -1
31 25 28 3 0 -1 -28 25
31 27 22 19 1 -3 -5 17
31 27 26 13 2 -5 -4 11
31 27 29 2 0 -1 -29 27
31 29 26 15 1 -5 -7 5
31 29 30 1 0 -1 -30 29
31 3 17 14 0 -1 -17 3
31 3 22 1 1 1 -9 5
31 5 18 13 0 -1 -18 5
31 5 22 3 1 3 -5 3
31 7 19 12 0 -1 -19 7
31 7 21 8 1 1 -10 9
31 9 20 11 0 -1 -20 9
32 10 21 11 0 -1 -21 10
32 14 21 13 1 -12 -2 1
32 14 23 9 0 -1 -23 14
32 18 25 7 0 -1 -25 18
32 2 17 15 0 -1 -17 2
32 22 23 15 1 -6 -5 3
32 22 27 5 0 -1 -27 22
32 26 25 15 1 1 -7 -4
32 26 27 11 1 -8 -4 3
32 26 29 3 0 -1 -29 26
32 30 29 11 1 -7 -5 4
32 30 31 1 0 -1 -31 30
32 6 19 13 0 -1 -19 6
32 6 23 1 1 1 -9 4
33 1 17 16 0 -1 -17 1
33 1 23 4 1 1 -10 7
33 13 23 10 0 -1 -23 13
33 13 25 2 1 3 -6 5
33 17 20 17 2 -5 -2 9
33 17 25 8 0 -1 -25 17
33 19 23 14 2 3 -3 -1
33 19 25 10 1 1 -8 -1
33 19 26 7 0 -1 -26 19
33 23 28 5 0 -1 -28 23
33 25 29 4 0 -1 -29 25
33 29 26 17 1 1 -7 -5
33 29 31 2 0 -1 -31 29
33 31 25 20 1 -3 -6 19
33 31 31 8 3 1 -4 -1
33 31 32 1 0 -1 -32 31
33 5 19 14 0 -1 -19 5
33 7 20 13 0 -1 -20 7
34 0 23 7 1 3 -5 2
34 12 19 17 1 -8 -3 1
34 12 23 11 0 -1 -23 12
34 12 25 5 1 1 -9 2
34 16 25 9 0 -1 -25 16
34 20 27 7 0 -1 -27 20
34 24 29 5 0 -1 -29 24
34 28 23 21 1 -4 -9 5
34 28 31 3 0 -1 -31 28
34 32 27 19 1 1 -7 -6
34 32 33 1 0 -1 -33 32
34 4 19 15 0 -1 -19 4
34 8 21 13 0 -1 -21 8
34 8 23 9 1 1 -11 10
35 11 23 12 0 -1 -23 11
35 1 18 17 0 -1 -18 1
35 13 21 16 1 3 -4 -1
35 13 24 11 0 -1 -24 13
35 15 23 14 1 -13 -2 1
35 15 26 7 1 1 -9 1
35 17 26 9 0 -1 -26 17
35 19 27 8 0 -1 -27 19
35 19 28 3 2 1 -6 1
35 23 29 6 0 -1 -29 23
35 25 22 21 1 1 -13 17
35 25 27 14 1 1 -8 -3
35 27 31 4 0 -1 -31 27
35 29 32 3 0 -1 -32 29
35 31 33 2 0 -1 -33 31
35 3 19 16 0 -1 -19 3
35 33 31 14 2 -5 -5 13
35 33 34 1 0 -1 -34 33
35 5 24 7 1 1 -11 9
35 9 22 13 0 -1 -22 9
36 10 23 13 0 -1 -23 10
36 14 25 11 0 -1 -25 14
36 2 19 17 0 -1 -19 2
36 22 23 19 1 1 -13 16
36 22 29 7 0 -1 -29 22
36 2 23 11 1 4 -4 1
36 2 25 5 1 1 -11 8
36 26 25 19 1 -5 -7 4
36 26 31 5 0 -1 -31 26
36 34 35 1 0 -1 -35 34
37 11 24 13 0 -1 -24 11
37 11 27 4 1 1 -10 3
37 1 19 18 0 -1 -19 1
37 1 26 3 1 1 -11 7
37 13 25 12 0 -1 -25 13
37 15 26 11 0 -1 -26 15
37 17 27 10 0 -1 -27 17
37 19 24 17 1 1 -13 15
37 19 28 9 0 -1 -28 19
37 21 28 11 1 1 -9 -1
37 21 29 8 0 -1 -29 21
37 23 25 18 2 -5 -3 11
37 23 30 7 0 -1 -30 23
37 25 31 6 0 -1 -31 25
37 27 32 5 0 -1 -32 27
37 29 24 23 1 -3 -5 19
37 29 31 12 1 -9 -4 3
37 29 32 9 2 1 -6 -1
37 29 33 4 0 -1 -33 29
37 31 29 18 1 1 -8 -5
37 31 34 3 0 -1 -34 31
37 3 20 17 0 -1 -20 3
37 3 25 8 2 1 -7 5
37 33 35 2 0 -1 -35 33
37 35 36 1 0 -1 -36 35
37 37 35 12 1 -7 -6 5
37 5 21 16 0 -1 -21 5
37 5 24 11 1 3 -5 1
37 7 22 15 0 -1 -22 7
37 9 23 14 0 -1 -23 9
37 9 25 10 1 1 -12 11
37 9 26 7 2 3 -4 1
38 12 25 13 0 -1 -25 12
38 16 25 15 1 -14 -2 1
38 16 27 11 0 -1 -27 16
38 16 29 3 1 3 -7 6
38 20 29 9 0 -1 -29 20
38 24 29 13 1 -11 -3 2
38 24 31 7 0 -1 -31 24
38 28 33 5 0 -1 -33 28
38 32 35 3 0 -1 -35 32
38 36 29 23 1 -4 -11 7
38 36 37 1 0 -1 -37 36
38 4 21 17 0 -1 -21 4
38 4 27 1 1 1 -11 6
38 8 23 15 0 -1 -23 8
38 8 27 5 1 4 -5 3
39 11 25 14 0 -1 -25 11
39 1 20 19 0 -1 -20 1
39 13 22 19 1 -9 -3 1
39 13 29 2 2 1 -7 3
39 17 28 11 0 -1 -28 17
39 17 29 8 1 1 -10 1
39 19 29 10 0 -1 -29 19
39 23 25 20 1 1 -14 17
39 23 31 8 0 -1 -31 23
39 23 32 1 3 1 -5 1
39 25 28 17 1 -7 -5 3
39 25 32 7 0 -1 -32 25
39 29 34 5 0 -1 -34 29
39 31 29 20 1 -5 -8 5
39 31 35 4 0 -1 -35 31
39 35 37 2 0 -1 -37 35
39 37 31 22 1 1 -8 -7
39 37 34 17 1 3 -9 11
39 37 38 1 0 -1 -38 37
39 5 22 17 0 -1 -22 5
39 7 23 16 0 -1 -23 7
39 7 28 1 1 1 -11 5
40 10 27 11 1 1 -13 12
40 10 29 3 1 1 -11 4
40 14 27 13 0 -1 -27 14
40 18 29 11 0 -1 -29 18
40 18 31 1 1 4 -6 5
40 2 21 19 0 -1 -21 2
40 22 31 9 0 -1 -31 22
40 26 33 7 0 -1 -33 26
40 30 31 17 1 1 -9 -4
40 34 33 17 1 -6 -7 5
40 34 37 3 0 -1 -37 34
40 38 39 1 0 -1 -39 38
40 6 23 17 0 -1 -23 6
41 11 26 15 0 -1 -26 11
41 11 30 1 1 3 -7 5
41 1 21 20 0 -1 -21 1
41 1 29 0 2 3 -5 3
41 13 22 21 1 5 -3 -1
41 13 27 14 0 -1 -27 13
41 13 30 5 1 1 -11 3
41 15 28 13 0 -1 -28 15
41 17 27 16 1 -15 -2 1
41 17 29 12 0 -1 -29 17
41 19 30 11 0 -1 -30 19
41 21 31 10 0 -1 -31 21
41 23 24 23 1 -5 -7 3
41 23 31 12 1 1 -10 -1
41 23 32 9 0 -1 -32 23
41 23 33 4 2 1 -7 1
41 25 33 8 0 -1 -33 25
41 27 26 23 1 1 -15 19
41 27 34 7 0 -1 -34 27
41 29 30 19 2 -5 -4 13
41 29 35 6 0 -1 -35 29
41 31 36 5 0 -1 -36 31
41 3 22 19 0 -1 -22 3
41 3 26 13 1 7 -3 1
41 3 29 2 1 1 -12 7
41 33 32 19 1 1 -9 -5
41 33 37 4 0 -1 -37 33
41 35 38 3 0 -1 -38 35
41 37 30 25 1 -3 -7 23
41 37 38 9 3 1 -5 -1
41 37 39 2 0 -1 -39 37
41 39 40 1 0 -1 -40 39
41 41 40 9 4 -9 -4 9 RECORD P 4
41 5 23 18 0 -1 -23 5
41 7 24 17 0 -1 -24 7
41 7 28 9 1 1 -13 11
41 9 25 16 0 -1 -25 9
42 16 29 13 0 -1 -29 16
42 16 31 7 1 1 -11 2
42 20 31 11 0 -1 -31 20
42 32 35 13 1 -10 -4 3
42 32 37 5 0 -1 -37 32
42 40 29 29 2 -7 -7 4
42 40 41 1 0 -1 -41 40
42 4 23 19 0 -1 -23 4
42 4 29 7 1 1 -13 10
42 8 25 17 0 -1 -25 8
43 11 27 16 0 -1 -27 11
43 11 29 12 1 1 -14 13
43 1 22 21 0 -1 -22 1
43 1 27 14 1 5 -4 1
43 1 30 5 1 1 -13 9
43 13 28 15 0 -1 -28 13
43 15 26 19 1 3 -5 -1
43 15 29 14 0 -1 -29 15
43 17 30 13 0 -1 -30 17
43 19 24 23 2 -5 -2 11
43 19 31 12 0 -1 -31 19
43 19 32 9 1 1 -11 1
43 19 33 4 1 3 -8 7
43 21 28 19 1 1 -15 17
43 21 32 11 0 -1 -32 21
43 23 30 17 2 3 -4 -1
43 23 33 10 0 -1 -33 23
43 25 34 9 0 -1 -34 25
43 27 35 8 0 -1 -35 27
43 29 33 16 1 1 -10 -3
43 29 36 7 0 -1 -36 29
43 31 27 26 1 1 -16 21
43 31 37 6 0 -1 -37 31
43 3 23 20 0 -1 -23 3
43 33 37 10 2 1 -7 -1
43 33 38 5 0 -1 -38 33
43 35 31 24 2 -9 -5 3 RECORD P 2
43 35 39 4 0 -1 -39 35
43 37 40 3 0 -1 -40 37
43 39 34 23 1 1 -9 -7
43 39 41 2 0 -1 -41 39
43 41 33 26 1 -3 -8 25
43 41 42 1 0 -1 -42 41
43 5 24 19 0 -1 -24 5
43 7 25 18 0 -1 -25 7
43 7 30 7 2 1 -8 5
43 9 26 17 0 -1 -26 9
43 9 31 2 1 1 -12 5
44 10 27 17 0 -1 -27 10
44 14 25 21 1 -10 -3 1
44 14 29 15 0 -1 -29 14
44 18 29 17 1 -16 -2 1 RECORD P 1
44 18 31 13 0 -1 -31 18
44 2 23 21 0 -1 -23 2
44 2 31 3 1 1 -13 8
44 26 35 9 0 -1 -35 26
44 30 37 7 0 -1 -37 30
44 34 39 5 0 -1 -39 34
44 38 31 27 1 -4 -12 7
44 38 39 13 1 -9 -5 4
44 38 41 3 0 -1 -41 38
44 42 35 25 1 1 -9 -8
44 42 41 13 1 -8 -6 5
44 42 43 1 0 -1 -43 42
44 6 25 19 0 -1 -25 6
44 6 31 5 1 3 -7 4
45 11 28 17 0 -1 -28 11
45 11 32 7 1 5 -5 3
45 1 23 22 0 -1 -23 1
45 13 29 16 0 -1 -29 13
45 17 31 14 0 -1 -31 17
45 17 34 1 2 1 -8 3
45 19 32 13 0 -1 -32 19
45 23 34 11 0 -1 -34 23
45 25 29 22 1 1 -16 19
45 25 34 13 1 1 -11 -1
45 29 37 8 0 -1 -37 29
45 31 38 7 0 -1 -38 31
45 35 29 28 1 -3 -6 23
45 35 37 16 3 -7 -4 11 RECORD P 3
45 37 41 4 0 -1 -41 37
45 41 37 22 1 -5 -10 7
45 41 43 2 0 -1 -43 41
45 43 41 16 2 1 -7 -3
45 43 44 1 0 -1 -44 43 RECORD ABS 44
45 5 31 8 1 1 -14 11
45 5 32 1 1 1 -13 7
45 7 26 19 0 -1 -26 7
45 7 29 14 1 3 -6 1
|x| |y| |u| |v| p q r s
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This method is based on Fricke and Klein (1897). I have a cheap reprint of the German original, an English translation has just come out. This is the first volume on automorphic forms, the one that treats discrete groups.
Part One: all integer solutions of $$ ab = 2cd $$ are parametrized (up to change of order) by four integers $p,q,r,s$ and $$ a = 2pr, \; b = qs, \; c = qr, \; d = ps \; . $$ In order to get abcd coprime we need $\gcd(p,q) = 1$ and $\gcd(r,s) = 1.$
FRIDAY: actually had a good program for finding matrix, this one is more attractive... Take column vector $$ \left( \begin{array}{c} a \\ b \\ c \\ d \\ \end{array} \right) $$ and take the matrix product (with abcd in the various orders, especially bacd) $$ \left( \begin{array}{cccc} 1 & 1 & 2 & 0 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ \end{array} \right) \left( \begin{array}{c} a \\ b \\ c \\ d \\ \end{array} \right) = \left( \begin{array}{c} x \\ y \\ u \\ v \\ \end{array} \right) $$ and throw out any where $\gcd(x,y,u,v) \neq 1.$
Alright, wrote them out without taking absolute values, $$ \color{magenta}{ x = 2pr+qs+2qr \; , \; \; y = 2pr + qs + 2ps \; , \; \; u=2pr +qr +ps \; , \; \; v = qs +qr+ps \; \; .} $$
$$ \left( \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 2 & 0 & 1 & 1 \\ 0 & 2 & 1 & 1 \\ \end{array} \right) \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -2 & 0 \\ 0 & 0 & 0 & -2 \\ \end{array} \right) \left( \begin{array}{cccc} 1 & 1 & 2 & 0 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ \end{array} \right) = \left( \begin{array}{cccc} 0 & 2 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & -4 \\ 0 & 0 & -4 & 0 \\ \end{array} \right) $$
Did some experiments, this is very successful... if we simply take the absolute values of xyuv, the order does not seem to matter at all. Also, it is not quite possible to demand my $p$ to be zero, this is still a four variable parametrization, but it may be taken quite small..
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