Test this out and see if you can find the flaw in the program
Clear[a, b, p]; k = 2^3217 - 1; p =
Solve[{(2 (1 + 2 k) (-k + a (1 + k - b)) b)/((-1 + a b) (2 k +
a b) (1 + 2 k + a b)) == 0, a > 0, b > 0}, {a, b},
Integers]; p = FromDigits[Differences[Flatten[{a, b} /. p]]]; If[
p == 0, Print[k, " <-Is Prime"], Print[k, " <- Is Not Prime"]]
try some values for k.