Let $p=x^2+11x-1=1\pmod 5$ be a prime. Show that $x$ is a quintic residue $\pmod p$. It holds for $x<200$ and should hold for all such $x$. Any proof ideas? Thanks in advance.

  • 2
    $\begingroup$ I imagine you'd want to look at how the 5th cyclotomic polynomial ($(x^5-1)/(x-1)$) might factor. I'm guessing, though. $\endgroup$
    – user14972
    Sep 19, 2018 at 23:00

1 Answer 1


Here's a more general result . . .

Let $x$ be an integer, and let $n=x^2+11x-1$.

Claim:$\;$If $n$ is not a multiple of $5$, then $x$ is a $5$-th power, mod $n$.


Let $a$ be an integer such that $5a\equiv 1\;(\text{mod}\;n)$, and let $w=a(x+3)$. \begin{align*} \text{Then}\;\;w^5&=\bigl(a(x+3)\bigr)^5\\[4pt] &\equiv (3125a^5)x\;(\text{mod}\;n)\;\;\;\text{[by polynomial long division]}\\[4pt] &\equiv \bigl((5a)^5\bigr)x\;(\text{mod}\;n)\\[4pt] &\equiv x\;(\text{mod}\;n)\\[4pt] \end{align*} which proves the claim.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.