Sorry, I can't follow your argumentation, at least (MP+2)/3 isn't prime for MP10,11, 13 .. 22
for instance: (MP10+2)/3= (2^89+2)/3= 206323339880896712483187371 =179*62020897*18584774046020617 q.v.: http://www.research.att.com/~njas/sequences/A107290 Andreas Stiller ----- Original Message ----- From: <[email protected]> To: <[email protected]> Sent: Sunday, October 25, 2009 8:00 PM Subject: Prime Digest, Vol 60, Issue 4 > Send Prime mailing list submissions to > [email protected] > > To subscribe or unsubscribe via the World Wide Web, visit > http://hogranch.com/mailman/listinfo/prime > or, via email, send a message with subject or body 'help' to > [email protected] > > You can reach the person managing the list at > [email protected] > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of Prime digest..." > > > Today's Topics: > > 1. Wagstaff numbers and generalized form for Mersenne numbers > (mario turco) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Sat, 24 Oct 2009 07:40:45 +0200 > From: "mario turco" <[email protected]> > Subject: [Prime] Wagstaff numbers and generalized form for Mersenne > numbers > To: <[email protected]> > Message-ID: <[email protected]> > Content-Type: text/plain; charset="iso-8859-1" > > 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 > [email protected] > http://hogranch.com/mailman/listinfo/prime > > > End of Prime Digest, Vol 60, Issue 4 > ************************************ > > _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
