IMO Shortlist 1978 problem 17
Dodao/la:
arhiva2. travnja 2012. Prove that for any positive integers
![x, y, z](/media/m/e/1/6/e160f3439547ca8c1afcc35a1c26f080.png)
with
![xy-z^2 = 1](/media/m/e/2/a/e2ad651d132ae0b2f1bda9aa90292b59.png)
one can find non-negative integers
![a, b, c, d](/media/m/a/b/a/aba147d136d904768670353792ec9289.png)
such that
![x = a^2 + b^2, y = c^2 + d^2, z = ac + bd](/media/m/b/3/5/b352c56c566f1dfec326987570a6ded1.png)
.
Set
![z = (2q)!](/media/m/5/f/6/5f63e93d870985c35bc9c0de291e1272.png)
to deduce that for any prime number
![p = 4q + 1](/media/m/e/c/2/ec21e48d3827a8c54fb20b274becd0b3.png)
,
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
can be represented as the sum of squares of two integers.
%V0
Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$.
Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.
Izvor: Međunarodna matematička olimpijada, shortlist 1978