hard or easy ? more, please...

[Jim Nastos]

On 4/25/08, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:

> I look for pairs of primes {p,q} such that
> (p+1)(p+2)=-1 (mod q)
> (q+1)(q+2)=-1 (mod p)
>
> Sorry for posting such uninteresting and/or trivial stuff, but if you
> happen to know the solution or any reference, thanks in advance.

  Is there any significance to these p,q s? (That is, is there any
reason you really want these to be primes?)
  With the same relationship but allowing {p,q} to be any pair of odd
numbers, you only seem to come across a couple of more pairs:
{3,3} (also solves your prime case)
{3,21}
{7,73}
{21, 507}
{73, 793}
{793, 8647}

What's interesting about this are the sequence of pairs:
{3,3} {3,21} {21,507} ... ?
{7,73} {73, 793}, {793, 8647} ... ?

And maybe you can use these to search the OEIS for some relevant
properties of your system of congruencies.

JN