Cfrgtkky

If a prime can be expressed as sum of two squares, then prove that the representation is unique.

My attempt:

If $a^2+b^2=p$, then it is obvious that $a,b$ of different parity.

Now, I assume the contraposition that the representation is not unique, $p=a^2+b^2=c^2+d^2$. Again, $c,d$ are of different parity.

Now, let $b,d$ be even and $a,c$ be odd.

So, $a^2+b^2=c^2+d^2 implies a^2-c^2=d^2-b^2 implies (a+c)(a-c)=(d-b)(d+b)$.

I cannot proceed any further. Please help.

The slickest way is via a little Algebraic Number Theory. If $p=a^2+b^2=c^2+d^2$ then $$p=(a+bi)(a-bi)=(c+di)(c-di)$$ Now ${bf Z}[i]$ is a unique factorization domain, so these two factorizations of $p$ show that $a+bi$ can't be a prime in ${bf Z}[i]$. We must have a non-trivial factorization $a+bi=(s+ti)(u+vi)$, whence $a-bi=(s-ti)(u-vi)$, and then $$p=(s^2+t^2)(u^2+v^2)$$ contradicting primality of $p$.

There are ways to answer your question without these advanced concepts, but I can never remember how it's done. I'm sure someone else will.

Here's an answer without Algebraic Number Theory. I found it in Shanks, Solved and Unsolved Problems in Number Theory.

Assume $$p=a^2+b^2=c^2+d^2tag1$$ with all variables positive integers. Then $$p^2=(a^2+b^2)(c^2+d^2)=a^2c^2+a^2d^2+b^2c^2+b^2d^2$$ and you can verify by just multiplying everything out that $$p^2=(ac+bd)^2+(ad-bc)^2tag2$$ and $$p^2=(ac-bd)^2+(ad+bc)^2tag3$$ By (1), we have $$(p-a^2)d^2=(p-c^2)b^2$$ which implies $$p(d^2-b^2)=(ad-bc)(ad+bc)tag4$$ From (4), $p$ divides $ad-bc$, or $p$ divides $ad+bc$. If $p$ divides $ad-bc$, then from (2) we get $ad-bc=0$, so $d^2-b^2=0$, so $b=d$. If $p$ divides $ad+bc$, then from (3) we get $ac=bd$. Now $a$ and $b$ are relatively prime, so $a$ divides $d$, and $b$ divides $c$. Then by (1) we have $a=d$, and we have proved that the two representations of $p$ are the same.

This is probably something like what @Konstantinos was getting at in his answer.

The problem is trivialized if you are given that $x^2+y^2=n$ has $r_2(n)=4sum_{d|n} sin(pi d/2) $ solutions in the integers.

The number of ways $r_2(p)=4sum_{d|p} sin(frac{pi d}{2})=4(1+sin(pi p/2))$.

If $sin(pi p /2)=1$ then there are 8 solution in the integers. Namely, $(pm a,pm b)$ and $(pm b, pm a)$ satisfy $x^2+y^2=p$ where $aneq b$. There is another way one might imagine $8$ solutions: $(0,pm a),(pm a,0)$ and $(pm b,pm b)$ but this cannot happen because it would mean that $p$ is a perfect square and therefore not prime.

If $sin(pi p /2)=-1$ then we cannot write $p$ as the sum of squares.

$If sin(pi p /2)=0$ then $p=2$.

If $p=a^2+b^2=c^2+d^2$ then $$p=frac{(ac+bd)(ac-bd)}{(a+d)(a-d)}$$