1. Sum of squares

Can someone give me a hint for how to show if $p \equiv 1 \mod{4}$ then $p = a^2+b^2$?

Thanks a lot!

2. Originally Posted by chiph588@
Can someone give me a hint for how to show if $p \equiv 1 \mod{4}$ then $p = a^2+b^2$?

Thanks a lot!
Consider, $\mathbb{Z}[i]$. This is a principal ideal domain. Now the equation $x^2 \equiv -1(\bmod p)$ is solvable if $p\equiv 1(\bmod 4)$. Therefore, $p|(x^2+1)$ for some $x\in \mathbb{Z}$. If $p$ was irreducible then $p$ would also be prime (since the integral domain is a PID and so "irreducible" is equivalent to "prime") and so $p|(x+i)(x-i)\implies p|(x+i) \text{ or }p|(x-i)$. But this is impossible. Therefore, $p$ is reducible and $p=\alpha \beta$ where $\alpha,\beta \in \mathbb{Z}[i]$ and $\alpha,\beta$ are non-units. This means $p^2 = Np = (N\alpha)(N\beta)$ (norm). However, $N\alpha, N\beta \not = 1$ since they are not units and so $N\alpha = N\beta = p$. Therefore, if $\alpha = a+bi$ then it means $N\alpha = p \implies a^2 + b^2 = p$.

Additional problem: Prove that if $p=a^2+b^2$ and $p=c^2+d^2$ where $a,c$ are odd and $b,d$ are even and $a,b,c,d>0$ then $a=c\text{ and }b=d$. In other words, we have a certain level of uniqueness in the decomposition of $p$ into sum of squares.

3. Ah yes, I was having trouble seeing why p is reducible.

4. Here is a very elementary way of proving this result. Remember if $p\equiv 1(\bmod 4)$ then there is $a$ so that $a^2 \equiv -1(\bmod p)$. There is an elementary way of showing this too, just let $a = \left( \tfrac{p-1}{2} \right)!$ and use Wilson's theorem. But whatever, we will accept this result.

We need a Lemma first: let $(b,p)=1$ then there exists $x,y$ with $0 < |x|,|y| < \sqrt{p}$ so that $bx\equiv y(\bmod p)$.

To prove this let $n = [\sqrt{p}]+1$. And define $S = \{bx - y | 0\leq x,y \leq n-1\}$. The set $S$ has at most $n^2$ different values. Since $n^2 > p$ and there are only two $p$ residue classes of mod $p$ it means there are $bx_1 - y_1\text{ and }bx_2 - y_2$ so that $bx_1 - y_1 \equiv bx_2 - y_2(\bmod p)$ and $x_1\not = x_2$ or $y_1\not = y_2$ (by pigeonhole principle). If $x_1 = x_2$ then by considering this congruence it would mean $y_1 = y_2$, which is a contradiction. Similarly if $y_1 = y_2$ then it would force $x_1=x_2$, which is a contradiction. Therefore, $x_1\not =x_2 , ~ y_1\not = y_2$. Thus, we have $b(x_1 - x_2) \equiv (y_1 - y_2)(\bmod p)$. Set $x=x_1-x_2, ~ y=y_1-y_2$ and we see that $0 < |x|,|y| \leq n-1 < \sqrt{p}$. Remember that $x^2,y^2 < p \implies x^2+y^2 < 2p$, thus $k$ must be equal to $1$.

Now we can prove that $p$ is a sum of two squares. Let $a$ be a square root of $-1$ mod $p$ i.e. in less fancy language $a^2\equiv -1(\bmod p)$. Pick integers $x,y$ so that $ax\equiv y(\bmod p)$ and $0 < |x|,|y| < \sqrt{p}$. Therefore, $-x^2 \equiv a^2x^2 = (ax)^2 \equiv y^2(\bmod p)$. Thus, $p|(x^2+y^2) \implies x^2+y^2 = pk$ for some $k\in \mathbb{Z}^+$. We will complete the proof if we can show $k=1$.