Proth theorem extended

[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^((Q-1)/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 'Fermat-number' proofs by Proth along with my replacing 'm' with 'm+1'].

now, if 'p' is any prime divisor of 'R', then a^((Q-1)/4) = (a^k)^(2^(n-2)) == +/-1(mod p) implies that p == +/-1 (mod 2^n). thus, if 'R' is compo-site, '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^((Q-1)/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

[R. Gerbicz]

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^((Q-1)/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 counter-example in Pari-Gp. Here it is one:

Q=8355841, so n=15, k=255, and let a=3, the conditions are true.
a^((Q-1)/4)==1 mod Q, but Q is composite: Q=13*41*61*257

[10metreh]

I vote Misc. Math.

[retina]

So where is the error in the "proof"? I think it would be quite instructive to see where it goes wrong.

[R. Gerbicz]

Originally Posted by retina

So where is the error in the "proof"? I think it would be quite instructive to see where it goes wrong.

Yes, this is far from a standard proof in math. Here it is a big mistake:
"now, if 'p' is any prime divisor of 'R', then a^((Q-1)/4) = (a^k)^(2^(n-2)) == +/-1(mod p) implies that p == +/-1 (mod 2^n)"
This is totally false.
And R=Q in the "proof", if you haven't observed it.