How many prime numbers $p$ are there such that
$ 29^p +1 $ is a multiple of $p$?

Question

mathmate · Answer

@eliesaab
I only found 2,3,5 up to p=100,000.
Would that be a good conjecture to assume ...

eliesaab · Answer

Yes, it is. Now you have to prove it

zarkon · Answer

Fermat's Little Theorem

eliesaab · Answer

Yes, Fermat's Little Theorem can give the proof. It is really a one line proof.

mathmate · Answer

@eliesaab
Would you share the proof with us?  I am rather weak on number theory.  
But there is no rush, in case you're hoping to see other comments, ideas, or proofs.

eliesaab · Answer

Here it is
If $ p = 29 $ we have
$$

29 \not | 29^p +1
$$
and hence p and 29 are relatively prime.
By Fermat's Little Theorem
$$
29^p+1 \equiv 29+1 \equiv 0 \text {  (mpd p) }
$$
It follows that p divides 30 and hence p=2,3,5

eliesaab · Answer

mod p

mathmate · Answer

@eliesaab Thank you!  It does look simple!  :)

eliesaab · Answer

you are welcome @mathmate