A ring of odd primes

steiner1745

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?

micromass

(2,5,13) also works...

agentredlum

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.