If $p~ \mid~ m^p + n^p$, prove $p~ \mid~ \frac{m^p + n^p}{m+n}$.

Question

If $p \mid m^p + n^p$, and $p$ is a prime greater than $2$, prove $$p \mid \frac{m^p + n^p}{m+n}.$$

No clue how to start. Clearly $p \mid m + n$, but then what. I feel very less information is given. Yet primes are full of surprises.

Thanks in advance.

This is an exact duplicate of a very recent question, but I can't sem to find the duplicate... — Caleb Stanford, May 07 '14 at 05:20
@Goos The "duplicate" mark is incorrect. This question implies the older question, but if $p^2\mid m+n$ then the original problem doesn't imply this one. — Thomas Andrews, May 07 '14 at 12:45

score 0 · Answer 1 · answered May 07 '14 at 05:12

0

You can add arbitrary terms of the form $pm^an^b$ to the right side, and it will remain true. Then try using binomial theorem.

answered May 07 '14 at 05:12

user2485211

score 0 · Accepted Answer · edited May 07 '14 at 06:10

0

Hint : see $m \equiv -n$ $(\text {mod} p)$. Now factorise the part $\dfrac{m^p + n^p}{m + n}$ and then apply $m \equiv -n$ $(\text {mod} p)$.

edited May 07 '14 at 06:10

William Hilbert

answered May 07 '14 at 05:19

soumajit das

2 Answers2