# nt.number theory – find all \$ m \$ such \$ 2 ^ m + 1 | 5 ^ m-1 \$

The problem comes from a problem I found when I wrote the article

Find all the whole numbers $$m$$ such
$$2 ^ {m} +1 | 5 ^ m-1$$
It seems that there is no solution, I think it might be necessary to use the knowledge of quadratic reciprocity to solve this problem.
Yes $$m$$ it's strange then $$2 ^ m + 1$$ It is divisible by 3 but $$5 ^ m-1$$ It is not.
so $$m$$ be even take $$m = 2n$$,so
$$4 ^ n + 1 | 25 ^ m-1$$
Yes $$n$$ it's strange, then $$4 ^ n + 1$$ It is divisible by $$5$$,but $$25 ^ m-1$$ It is not
so $$n$$ it's even, take $$n = 2p$$,we have
$$16 ^ p + 1 | 625 ^ m-1$$