Yes, but In your example 206323339880896712483187371 isn't a prime number.
(MP10+2)/3= (2^89+2)/3= 206323339880896712483187371 =179*62020897*18584774046020617 Theorem A (Rosario Turco):Necessary and sufficient condition so that Mp = 2 ^ p-1 is a prime number is that all three conditions are true:1) p is prime and of any form (4k +1 or 4k +3)2) If p = 4k +3 then S = 2p +1 mustn't be a prime number3) Mp +2 = 3 * d with d prime numberWhy Theorem A is true? We know Mp + 2 = p2 * p3 in general, with Mp = 2 ^ p1-1Which is equivalent to the fact that:(2 ^ p1 +1) / p2 = p3Now 2 ^ p1 +1 is a prime number if p1 = 2k; but if p1 is a prime number then 2 ^ p1 +1 is composite. Ex: 2 ^ 3 +1 = 9 2 ^ 5 +1 = 33 2 ^ 7 +1 = 129 etcSo it must be p1=4k+1 or p1=4k+3 (but S=2p1+1 mustn't be a prime number):p1=4k+1 with k even but odd power of 2 (k=2^1, k=2^3 etc) es: 4*2+1=9, 4*2^3+1=33p1=4k+3 with k>1 and odd, multiple of 3 (k=3,6,9 etc)So a divisor of 2 ^ p1 +1 is p2 = 3; infact each (2 ^ p1 +1)% 9 = 3 or 6 if p1>3.If p2 = 3, p3 is a prime number? Yes, because it is also true that(2 ^ p1 +1) / p3 = p2. Rosario Turco _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
