I have new and interesting results on Mersenne numbers. It can also be used for the primality My article is Possiamo usare diversamente i numeri di Wagstaff?
Wagstaff numbers have the potential still unexplored, and their definition is: W = (2 ^ p +1) / 3 (1) In (1) there are several things to highlight: 1) W is first 2) 3 is first 3) 1 is coprime 4) p is prime Generalizing we are working, then, with expressions like: 2 ^ x = p1 + p2 * p3 ^ 2 or p1 = p2 * p3 - x (2) where p1, p2, p3, and x are first In (2) p2, compared to (1), has the role of 3, p3, the role of the role of 1 W ex. Suppose at x = 1 to (2) leads to these conclusions: p1 = log2 (p2-p3 * x) (3) Then from (3) for a given P1 (because there is assigned the Mersenne number 2 ^ p1-1), if p2 is fixed (the denominator in (1)) and if x = 1, then by varying p3 we obtain the equality (3). In particular, with x = 1 the largest number of equalities is obtained for p2 = 3 Another way to exploit the (2) is, if x = 1: Mp = 2 ^ p1-1 = p2 * p3-2 (4) Mp +2 = p2 * p3 (5) The (5) says that Mp +2 is a composite number (semiprime) derived from the product of 2 primes (referred to by numbers instead Wagstaff know that p2 = 3). Theorem (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 shape (4k +1 or 4k +3) 2) If p = 4k +3 S = 2p +1 be non-prime 3) Mp +2 = 3 * p Ex: 2 ^ 3-1 = 7 Mersenne prime then 1) is true, the 2) also, for the 3) is: 7 +2 = 3 * 3 which Mp is prime 2 ^ p1 = p2 * p3 - x Interesting to see if x is not 1, so (2) becomes: Mp = 2 ^ p1 -1 = p2 * p3 - x - 1 (6) equivalent to: Mp + x +1 = p2 * p3 We will see that also serve two conditions: x +1 = p1-1 p2 = p1 or Mp+p1-1 = p1 * p3 (7) or Mp=p1*p3-(p1-1) With the generalization (7) was born Theorem (Rosario Turco): Necessary and sufficient condition so that Mp = 2 ^ (p1-1) is a prime number is that all three conditions: 1) p1 is a prime number and any shape (4k +1 or 4k +3) 2) If p1 = 4k +3 then S = 2p1 + 1 must be non-prime 3) Mp + p1-1 = p1 * p3 with p3 prime number Now we will see why (7) is true: If in (6) say for example: x = 3 Mp + 4 = p2 * p3 Ex: 2 ^ 3-1 = 7 Mersenne prime 7 +4 = 11 first 11 is impossible, because x +1 = 4 is not good 7 +2 = 3 * 3 OK!! 2 ^ 7-1 is prime. Note that in 3 * 3 is just the exponent p1 = 3 Mp = 2 ^ 3-1 Examples 2 ^ 5-1 = 31 31 + 4 = 35 = 5 * 7 = 35 then 31 is a Mersenne prime! Note that in 5 * 7 is precisely the exponent p1 = 5 Mp = 2 ^ 5-1 2 ^ 7-1 = 127 127 + 6 = 133 = 7 * 19 2 ^ 13-1 + 12 = 8191 +12 = 8203 = 13 * 631 Hence: x +1 = p1-1 The (6), without generalization has another interest. If p3 is chosen as a big prime number (eg 127) it would be to test the primality with initially smaller prime number or you can check that (Mp + p1-1) / p3 = p1 (8) Eg 2 ^ 13-1 = 12 + (8191 + 12) / 631 = 13. So I must search a divisor p3 of Mp+p1-1 such that the (8) is true. Rosario Turco _______________________________________________ Prime mailing list Prime@hogranch.com http://hogranch.com/mailman/listinfo/prime
