Proth theorem extended
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:
if 'm' is from the set of natural numbers, then every odd prime divisor 'q' of a^(2^(m+1))+/1 implies that q == +/1(mod a^(m+2)) [concluded from generalized 'Fermatnumber' proofs by Proth along with my replacing 'm' with 'm+1'].
now, if 'p' is any prime divisor of 'R', then a^((Q1)/4) = (a^k)^(2^(n2)) == +/1(mod p) implies that p == +/1 (mod 2^n). thus, if 'R' is composite, 'R' will be the product of at least two primes each of which has minimum value of (2^n +1), and it follows that...
k*2^n +1 >= (2^n +1)*(2^n +1) = (2^n)*(2^n) + 2*(2^n) +1; but the
1's cancel, so k*(2^n) >= (2^n)*(2^n) + 2*(2^n) and upon dividing by 2^n... k>= 2^n +2.
moreover, this result is incompatible with our definition, so if k<= 2^n +1 and a^((Q1)/4) == +/1(mod Q), then 'Q' is prime.
finally, if either k= 2*m or k= 2*m +1, then 'Q' is still 'odd'; Q= 2*m*2^n +1= m*2^(n+1)+1, or Q= (2m+1)*2^n +1= m*2^(n+1) +2^n+1.
*QED
