Quote:
Originally Posted by Bill Bouris
Proth theorem extended:
let Q= k*2^n +1, where n=>3 is a odd natural number & k<= 2^n +1. if for some 'a', a^((Q1)/4) == +/1(mod Q), then 'Q' is prime. 'k' doesn't need to be restricted to only 'odd' numbers, either.
proof:

When I see such a "theorem" I try to find a counterexample in PariGp. Here it is one:
Q=8355841, so n=15, k=255, and let a=3, the conditions are true.
a^((Q1)/4)==1 mod Q, but Q is composite: Q=13*41*61*257