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Let $x$ be an odd number, $y$ a positive integer, and $p$ an odd prime number such that $x^2+y^2=p$. Show that there exists an integer $a$ such that $\dfrac{a^2-x}{p}\in \mathbb{Z}.$

This problem is one that my friend asked me, and I've considered it for some time, but I can't prove it. Thank you everyone.

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  • $\begingroup$ i.e., $x$ needs to be Quadratic Residue $\pmod p$ $\endgroup$ May 19, 2013 at 7:46
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    $\begingroup$ Unsourced, unmotivated, no sign of any effort by OP beyond cut'n'paste. Voting to close. $\endgroup$ May 19, 2013 at 7:47
  • $\begingroup$ @GerryMyerson: We should let the OP respond. $\endgroup$
    – Inceptio
    May 19, 2013 at 7:48

2 Answers 2

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Using Jocobi symbol form of quadratic reciprocity, $$ \left(\frac{x}{p}\right) = \left(\frac{p}{x}\right)(-1)^{\tfrac{x-1}{2}\tfrac{p-1}{2}}=1. $$

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Hint: Because $p$ is odd and $p=x^2+y^2$ for some integers $x$ and $y$, we know that $p\equiv 1\bmod 4$. Let the prime factorization of $x$ be $$x=q_1^{e_1}\cdots q_r^{e_r}$$ (which includes no $2$'s, since $x$ is odd). Use the fact that $x^2\equiv 0\bmod q_i$ for all $i$, together with quadratic reciprocity, to prove that $\big(\!\frac{x}{p}\!\big)=1$, so that there is an integer $a$ with $a^2\equiv x\bmod p$.

Full argument in spoiler:

$$\begin{align*} \left(\frac{x}{p}\right)&=\left(\frac{q_1}{p}\right)^{e_1}\cdots\left(\frac{q_r}{p}\right)^{e_r}\\\\\\ &=\left(\frac{p}{q_1}\right)^{e_1}\cdots\left(\frac{p}{q_r}\right)^{e_r}\\\\\\ &=\left(\frac{x^2+y^2}{q_1}\right)^{e_1}\cdots\left(\frac{x^2+y^2}{q_r}\right)^{e_r}\\\\\\ &=\left(\frac{y^2}{q_1}\right)^{e_1}\cdots\left(\frac{y^2}{q_r}\right)^{e_r}\\\\\\ &=1^{e_1}\cdots 1^{e_r}\\\\\\ &=1 \end{align*}$$

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  • $\begingroup$ Thank you,But can you share your all answers, $\endgroup$
    – math110
    May 19, 2013 at 8:12
  • $\begingroup$ It's only been 4 minutes since I posted this hint. Keep trying to figure it out. "What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises." - G. C. Lichtenberg $\endgroup$ May 19, 2013 at 8:13
  • $\begingroup$ I just can't see it...and it is important that $\,x^2+y^2=p\,$ and not merely $\,x^2+y^2=0\pmod p\,$ , since for example $\,7^2+6^2=0\pmod {17}\,$ , yet $\,7\,$ is not a quadratic residue modulo $\,17\,$ . $\endgroup$
    – DonAntonio
    May 19, 2013 at 8:15

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