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Let $p$ be a prime number and $a,b,c$ integers such that $a$ and $b$ are not divisible by $p$. Prove that the equation $ax^2+by^2 \equiv c$ (mod $p$) has integer solutions.

I am trying to come up with a solution using Pigeonhole Principle, but I have limited knowledge of number theory. So I am having hard time coming up with an idea to start with.

Bill Dubuque
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Chanhyuk Park
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1 Answers1

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How limited is "very limited"? And why do you write the post in a command form ("Prove that...")? It is not how you normally go around asking questions in person, surely. That style of writing here is an obvious indication that this is homework.

Hint: Treat $p=2$ first, then let $p > 2$, rewrite the congruence as $ax^2 \equiv -by^2 + c \bmod p$, and count the number of values of both sides (your coefficients are fixed). How many squares mod $p$ are there (include $0$).

KCd
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  • It’s not a homework problem actually^^. I’m just studying outside of my class. – Chanhyuk Park Mar 30 '24 at 05:44
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    Ok. But when posting questions, don't write them by starting with "Show that..." or "Prove that..". It is more natural to tell us what you're doing, e.g., "I'm trying to show that...". Maybe even mention the book. – KCd Mar 30 '24 at 15:25