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A ring of odd primes

  1. Jul 2, 2011 #1
    I got this question from another
    forum and it's driving me crazy.
    Find all triples of odd primes,
    p,q,r such that
    p^2+1 is divisible by q, q^2+1 is divisible by r
    and r^2+1 is divisible by p.
    Two such triples are 5,13,17
    and 17,29,421. If we assume
    p<q<r, then there are no other
    such triples with p<10^7.
    Are there any others?
    Anyone have any ideas?
    From quadratic residue theory
    we know that p,q,r are all
    congruent to 1(mod 4).
    Can we say more?
    Last edited: Jul 2, 2011
  2. jcsd
  3. Jul 6, 2011 #2
    (2,5,13) also works...
  4. Jul 6, 2011 #3
    nice observation but 2 is not an odd prime.

    Years ago i thought i solved the BEAL CONJECTURE because i found 3^5 + 10^2 = 7^3

    Then my math prof. pointed out ALL exponents must be integers greater than 2.:smile:
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