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I've stumbled upon this and I was wondering if anyone here could come up with a simple proof:

Let $p$ be a prime such that $p\equiv 1 \bmod 8$, and let $a,b\geq 1$ such that $$p=a^2-2b^2.$$ Question: Is $b$ necessarily a square modulo $p$?

I have plenty of numerical data to support an affirmative answer, but the proof eludes me so far. For instance: \begin{align*} 17 & = 5^2 - 2\cdot 2^2\\ &= 7^2 - 2\cdot 4^2\\ & = 23^2 - 2\cdot 16^2\\ & = 37^2 - 2\cdot 26^2\\ & = 133^2 - 2\cdot 94^2\\ \end{align*} and $2\equiv 36$, $4$, $16$, $26\equiv 9$, $94\equiv 9 \bmod 17$ are squares.

Thanks!

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Try factoring: $(a+\sqrt{2}b)(a-\sqrt{2} b)\equiv 0 \mod p$ $\implies$ $a \equiv \sqrt{2} b \mod p$. If $a$ is an integer $\mod p$, then $\sqrt{2} b$ is an integer $\mod p$. Thus, can we conclude that $b$ is a square $\mod p$? –  rckrd Jun 7 '12 at 16:27
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I don't think one can simply work modulo $p$. Notice that $1^2-2\cdot 3^2\equiv 0 \bmod 17$, but $3$ is not a square mod $17$. –  Álvaro Lozano-Robledo Jun 7 '12 at 16:33
    
Can someone explain to me the fault my logic? $a=1$, $b=3$ doesn't satisfy $p \equiv 1 \mod 8$ as a counterexample, but I still feel unsure about my idea. –  rckrd Jun 7 '12 at 16:55
    
@Matt Do you want to work with negative primes, too? $$a^2-2b^2=1^2-2\cdot 3^2=-17$$ –  DonAntonio Jun 7 '12 at 16:59
    
@Matt Your logic is unclear. What is "an integer mod $p$"? And why is being "an integer mod $p$" imply being a square mod $p$? Also, when you say $a\equiv \sqrt{2}b\pmod p$, what exactly do you mean? (I am not saying that there is no meaning, but from your comment, it's not clear that you understand how to make sense of it.) –  Alex B. Jun 7 '12 at 16:59
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1 Answer

up vote 7 down vote accepted

Let $q$ be any prime divisor of $b$. Consider your equation modulo $q$. You will find that $p\equiv a^2 \pmod{q}$, so that $p$ is a square modulo $q$. By quadratic reciprocity (using that $p\equiv 1\pmod 4$), $q$ is a square modulo $p$. Since this is true for any prime divisor of $b$, $b$ is a square modulo $p$.

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+1 Beautiful, short, elegant. –  DonAntonio Jun 7 '12 at 17:02
    
Thank you. I am not sure how I missed this. I think it is an indication that I need a nap today. Thanks again! –  Álvaro Lozano-Robledo Jun 7 '12 at 17:24
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