Off the cuff speculation
Because of an alternative factoring algorithm I am working on,
the following speculation occurred to me.
I know that it is not wise to state a conjecture like this
before I can test it out myself.
Perhaps some of you might deem it a puzzle to see how
quickly you can find counter examples.
If z1 is an odd positive integer,
of the form p * 2**k + 1,
where p is a prime,
For c = at least one of the two solutions to
c**2 = z1 mod p**2, with 0 < c < p**2,
if z2 = c**2 + (z1 - c**2)/p**2
has factors z2 = x * y such that (y - x) = 2**k and x > 1,
then z1 is prime.