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Number theory; primes dividing proof

  1. Sep 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Show there exist infinitely many (p, q) pairs, (p ≠ q), s.t.
    p | 2[itex]^{q - 1}[/itex] - 1 and q | 2[itex]^{p - 1}[/itex] - 1

    2. Relevant equations

    We are allowed to assume that 2[itex]^{β}[/itex] - 1 is not a prime number or the power of a prime if β is prime.

    3. The attempt at a solution

    Using fermat's little theorem and the Chinese Remainder Theorem; we get

    pq | 2[itex]^{q - 1}[/itex] - 1 and pq | 2[itex]^{p - 1}[/itex] - 1,

    which together yield:

    pq | gcd(2[itex]^{q - 1}[/itex] - 1, 2[itex]^{p - 1}[/itex] - 1)

    which yields

    pq | 2[itex]^{gcd(q - 1, p - 1)}[/itex] - 1

    My strategy from here is to find some

    2[itex]^{α}[/itex] - 1 such that 2[itex]^{α}[/itex] - 1 | 2[itex]^{gcd(q - 1, p - 1)}[/itex] - 1

    and that pq | 2[itex]^{α}[/itex] - 1.

    However, this doesn't seem to be fruitful for me; I suspect I'm missing something very easy.

    Any hints would be greatly appreciated!

    Thanks!
     
    Last edited: Sep 29, 2012
  2. jcsd
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