Closed form of solutions of $xy-zv=±1$

Question

Let us consider a real dynamical system $s′=h(s)$. In order to study the stability of the central manifold, we reformulate the problem as follows: I am asking if this equation $$xy-zv=±1$$ has integer solutions $x,y,z,v$.

We can find some special cases, but I am interested in a closed-form.

$x,y$: any numbers. $z,v$: any divisors of $xy\pm1$. That's about as close to a closed form as it gets. — Ivan Neretin, Nov 13 '20 at 08:00

score 1 · Accepted Answer · 2020-11-13T17:46:08.743

1

By the Bezout identity, there are solutions iff $x,z$ are relative primes, and $y,v$ follow.

edited Nov 13 '20 at 17:46

answered Nov 13 '20 at 08:23

2

Unique modulo something, that is. – Ivan Neretin Nov 13 '20 at 08:50

Closed form of solutions of $xy-zv=±1$

1 Answers1