# Math Help - Primes of the form 4n+3

1. ## Primes of the form 4n+3

Let $p$ be a prime of the form $4n+3$. Prove that there exists integers $a,b$ such that $0 < a,b \leq p$ such that $p \mid (a^{2}+b^{2})$

2. Originally Posted by Chandru1
Let $p$ be a prime of the form $4n+3$. Prove that there exists integers $a,b$ such that $0 < a,b \leq p$ such that $p \mid (a^{2}+b^{2})$

As you put it the problem is trivial: choose $a,b=p\Longrightarrow p\mid a^2+b^2=2p^2$ , and if you meant $0 then the problem is false, as $p=3$ shows.

Tonio

3. Yeah, something makes me think that there are two mistakes in the statement of this problem! Because it is similar to the very interesting problem

Let $p$ be a prime of the form $4n+1$. Show that there exist integers $0 such that $p|a^2+b^2$.
which is not a trivial problem at all.