Thread: Primes of the form 4n+3

1. Primes of the form 4n+3

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

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

As you put it the problem is trivial: choose $\displaystyle a,b=p\Longrightarrow p\mid a^2+b^2=2p^2$ , and if you meant $\displaystyle 0<a,b<p$ then the problem is false, as $\displaystyle p=3$ 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 $\displaystyle p$ be a prime of the form $\displaystyle 4n+1$. Show that there exist integers $\displaystyle 0<a,b<p$ such that $\displaystyle p|a^2+b^2$.
which is not a trivial problem at all.