Condition for Quintic Reciprocity

Question

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.

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 $n$.

Proof:

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.