(In reply to
a computerized start but no proof and too small a sample size by Charlie)
I noticed that Q and P that you found both have recursive structure.
Q(n) = 254Q(n1)  Q(n2)
P(n) = 254P(n1)  P(n2)
The next solution of this form is
Q=39327734, P=2091028097, 2P+2=4162056196=64514^2
This could lead to an inductive proof that
P(n)^2 = 28Q(n)^2 + 1
and 2P(n) +2 is a perfect square
but I couldn't get it to come out.
This would only prove that numbers of this form work.
Other solutions beyond these could still satisfy P^2 = 28Q^2 +1

Posted by Jer
on 20100503 02:03:04 