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Show that for any prime p, there are integers $x, y$ such that $p\mid x^2+y^2+1$

Arnaldo
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user397055
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  • I've shown that. What now? Do you want me to present you the proof or something? – Wojowu Dec 07 '16 at 21:22
  • http://math.stackexchange.com/questions/1260928/show-that-for-any-prime-p-there-are-integers-x-and-y-such-that-p/1260948#1260948 – Elaqqad Dec 10 '16 at 15:38

1 Answers1

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Hint: Consider an odd prime $p$.

Show that the residues of $x^2$ modulo $p$ for $0\le x\le (p-1)/2$ are distinct.

Similarly,the residues of $-y^2-1$ modulo $p$ for $0\le y\le (p-1)/2$ are distinct. Then apply pigeonhole principle.

Q-Zhang
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