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This is the full question:

show that if $p$ is an odd prime then the number of ordered pair solutions of the congruence$x^2-y^2 \equiv a \pmod p,$ is $p-1$ unless $a \equiv 0 \pmod p,$ in which case number of solutions is $2p-1$.

Considering $x,y \,\in \mathbb{Z}$. In the second case, since $a \equiv 0 \pmod p,$ it follows that $p\mid (x-y)(x+y)$, but then there will be infinitely many solutions of this congruence relation because there are no bounds mentioned in the question on $x$ and $y$.

So is the question incomplete? or is it implicitly stated that $0\leq x,y<p$ .For this bound, do we get $2p-1$ solutions?

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    $\begingroup$ The congruence is modulo $p$. Therefore, there are only $p$ possible values that either $x$ or $y$ could be under this constraint. You could say that $2$ and $p+2$ are both values of $x$, but there is no distinction between them modulo $p$. Now suppose that $x$ is some non-zero value. How many values of $y$ solve the given congruence? What changes if $x$ is $0$? $\endgroup$
    – abiessu
    Aug 26, 2019 at 20:52
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    $\begingroup$ They are not asking for integer solutions but equivalence class solutions. There are only $p$ equivalence classes so there are only $p^2$ pairs of $(x,y)$ at all. $\endgroup$
    – fleablood
    Aug 26, 2019 at 21:09

2 Answers 2

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ETA: For whatever reason I thought the question here was to prove that the statement in the OP's thread in the shaded yellow box, which is what I did below.

You already answered your own question for the case where $a$ is zero so we now consider the case where $a$ is nonzero i.e., $a \in (\mathbb{F}_p)^{\times}$.

  1. Let $a \in (\mathbb{F}_p)^{\times}$. For every such nonzero $b \in (\mathbb{F}_p)^{\times}$, there is exactly one $c \in (\mathbb{F}_p)^{\times}$ satisfying $a=bc$.

  2. The above equation factors to $(x-y)(x+y) = a; x,y \in (\mathbb{F}_p)^{\times}$. By the above, for each nonzero $b=(x-y)\in (\mathbb{F}_p)^{\times}$, there is exactly one $c=(x+y)\in (\mathbb{F}_p)^{\times}$ such that $(x-y)(x+y)=a$.

  3. So for each nonzero $b\in (\mathbb{F}_p)^{\times}$, there is exactly one pair $(x,y); x,y \in (\mathbb{F}_p)^{\times}$ such that both $(x-y)=b$ and $(x-y)(x+y)=a$.

Can you finish from here.

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Trying to find integer solutions $0\le x,y< p$ of $x^2-y^2\equiv 0 mod p$ as you observed implicates that $p\mid (x-y)(x+y)$, thus you see $p\mid x+y$ or $p\mid x-y$ since $p$ is prime. For every $1\le x<p$ you have the solutions $(x,y)=(x,p-x)$ and $(x,y)=(x,x)$ that are in total $2p-2$ couples, and for $x=0$ the unic solution $(x,y)=(0,0)$. They are $2p-1$ total solution.

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