Brian Beesley wrote: > Date: Sun, 23 Nov 2003 08:53:55 +0000 > From: "Brian J. Beesley" <[EMAIL PROTECTED]> > Subject: Mersenne: Generalized Mersenne Numbers

> Define GM(a,b) = a^b-(a-1), so GM(2,b) = M(b); also GM(a,1) = 1 for all a What about GM(p,n)=p^n-(p-1)^p For p=2 we have GM(2,n)=2^n-1^2=M(n) For a prime p>2 we have GM(p,mp)=p^mp-(p-1)^p= (p^m)^p-(p-1)^p so it is divisible by p^m-(p-1) (for m=1 it equals 1). I have no idea how to prove (if it is possible to prove) that GM(p,n) is composite for composite n. However for p=3 we have (p-1)^p=2^3=8, so n should be greater than 1 and GM(3,2)=9-8= 1 (not prime, but not composite, too) GM(3,3)=27-8=19 prime GM(3,4)=81-8=73 prime GM(3,5)= 235= 5*47 GM(3,6)= 721=7*103 (as should be, see above) GM(3,7)= 2179 prime GM(3,8)= 6553 prime (for composite exponent!) GM(3,9) divisible by 3^3-2=25 GM(3,10), GM(3,11) composite GM(3,12) divisible by 3^4-2=79 GM(3,13) composite For p=5 we start from n=5 since 4^5=1024, whereas 5^4=625 GM(5,n) composite for n=5,6,8,9,10 (div. by 25-4=21) for n=7 and 11 -- prime GM(5,7)=77101 GM(5,11)=48827101 What happens if p is composite? [for p>3 we always start from n=p since in this case p^p > (p-1)^p and p^{p-1}<(p-1)^p, p=3 is the exception] GM(4,n) is composite for n<12 GM(6,11) is prime! 6^11-5^6=362797056-15625=362781431 Maybe it is caused by fact that p-1=5 is prime? What about more general meresennes MGM(p,q,n)=p^n-q^p, p,q prime (or not)? q^p is related to small Fermat theorem (I don't remember exactly how; probably q^(p-1) mod p =1 for prime p). As regards the analog of the LL test, I only remember that Lucas series is somehow related to Fibonacci series (see D.Knuth and others "Concrete mathematics"). Is similar series are related with a^b-(a-1) or a^b-(a-1)^a? Regards W Florek PS The most important: CONGRATULATIONS to Micheal, George and Scott and many, many others. WsF =============================================== Wojciech Florek (WsF) Adam Mickiewicz University, Faculty of Physics ul. Umultowska 85, 61-614 Poznan, Poland phone: (++48-61) 8295033 fax: (++48-61) 8295167 email: [EMAIL PROTECTED] _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers