Number Theory: Primes

Date: 07/10/2003 at 04:23:10
From: Carlos
Subject: Number theory

Find all primes p,q such that ((7^p-2^p)(7^q-2^q))/(pq) is an integer.

If p divides (7^p-2^p)(7^q-2^q), then p divides either (7^p-2^p) or 
(7^q-2^q) and vice versa. It can easily be deduced from Fermat's 
Little theorem that 7^p - 2^p = 5 (mod p). Thus p divides (7^p-2^p) 
iff p = 5.

From this I have figured that (p=5,q=5),(p=5,q=11),(p=5,q=61) are 
all solutions. Where I get stuck is for the case 7^q=2^q mod p and 
7^p=2^p mod q. I wonder if 7^q=2^q mod p and 7^p=2^p mod q have any 
solutions. I suspect that for p (not 5) to divide (7^q-2^q), p > q. 
This will prove that there are no other solutions. But I cannot 
prove this assertion.

Date: 07/10/2003 at 04:43:59
From: Doctor Jacques
Subject: Re: Number theory

Hi Carlos,

You are on the right track.

Assume that 7^p = 2^p (mod q) with p and q primes, and q <> 5. As q is 
obviously odd, 2 has an inverse mod q, namely:

  a = (q+1)/2

and we may write:

  (7a)^p = 1 (mod q)

Now, the order of (7a) mod q is either 1 (in which case q = 5) or p, 
and divides (q - 1), by Fermat's theorem. This should allow you to 
use your idea to conclude the proof.

Does this help?  Write back if you'd like to talk about this 
some more, or if you have any other questions.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/