Some days I should just keep it to myself. As some of you may have noticed, few odd primes are the sum of two other primes. (Not a good day for me, I think.) There may be some meat on this bone, tho'. >>Another thought: does the factor test already use Goldbach's conjecture? > Someone with more cleverness and time than I have right now might think > about using this fact: > > Given p,q integers and r = p+q. Then Mr = Mp.Mq + Mp + Mq How about p,q primes with r = p + 2q. This might work since Mq | M(2q) = Mq(Mq + 2). So, Mr = Mp.M(2q) + Mp + M(2q) The other notion of using a database of factors might still be interesting although the link to Goldbach's conjecture seems to vanish. > This is just: Mp.Mq = (2^p - 1)(2^q - 1) = 2^(p+q) - 2^p - 2^q + 1 = Mr - > Mp - Mq Suppose we want to factor test Mr. Perhaps we can (by trial and error) find primes p and q with r = p+2q. Mr, Mp and Mq are relatively prime in pairs, so none of the factors of Mp or Mq divide Mr. Now, if F is a test factor in the range, suppose we keep a database of Fp = Mp mod F. This is big, but once we have r=p+2q, we just calculate Fp.Fq.(Fq+2) + Fp +Fq(Fq+2) mod F. If 0, then F divides Mr. > One practical question is how bloody big is this collection of Fp's? > > Could this make test factoring run faster? > Joth _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
