Stern primes (A042978): A new higher lower bound

[Alonso Del Arte]

With a simple algorithm in Mathematica I was able in a matter of
seconds to confirm Jud McCranie's lower bound of 1299709 (that if
there is a Stern prime higher than 1493 it must be beyond the first
hundred thousand primes -- judging by the A number, A042978 was added
in 1999 or 2000).

Over a couple of hours (with the same algorithm) I've been able to
check up to 452930459 (the first 24 million primes). Like the problem
of oddsquarefree C(2n, n), this is one that seems like it should be
easy to prove why the highest value given in the table is the highest
value period, yet the proof remains elusive.

Laurent Hodges's paper says that as the prime q gets bigger there are
more ways to represent it as 2b^2 + p. My algorithm might be refined
by starting checking b closer to sqrt(p/2) rather than starting with b
= 1 if we're likelier to find an appropriate b and p there, but that's
a question I haven't yet explored in detail.

Merry Christmas!

Alonso