TITLE: Is 3 ever a seventh power mod a prime $p$ if $p\equiv 1 (7)$
QUESTION [5 upvotes]: I had a question asking when is 3  a seventh power modulo a prime $p$ if $p=1(7)$. However, I tried to find just one example using mathematica but I went up to primes in the thousands and I still couldn't find an example, so I began thinking this was a trick question. 
We were learning about quadratic reciprocity, but I wasn't sure how to extend what we learned about quadratic powers to seventh powers.

REPLY [3 votes]: There is much classical work on this and related questions. You can find pointers to the literature from the introduction of Stanislav Jakubec's $ $ Criterion for 3 to be eleventh power, $ $ Acta Mathematica et Informatica Universitatis Ostraviensis (1995), Vol. 03, 1,  excerpted below