Subject: [Prime] Re: You no longer need Lucas-Lehmer Double... A repeat of my proposal (now in email-format) :Given M8=2^31-1= 2 147 483 647 =A(0): Prove and certify the primality, so that all of us can check and verity at sight. Here is the proposal. 1) A(0-1) = 2 147 483 646 = 2 x 3 x 3 x 7x 11 x 31 x 151 x 331; B(0)=331 A(0 ) = 2 147 483 647 A(0+1) =2 147 483 648 = 2^31 (these prime factors are easily verifiable)
A(0 ) is prime, certified by A(0-1), A(0+1) and B(0) below. (One and the same algorithm must be used to factorize the sequential integers A(0-1) to A(0+!) 2) In case B(0 ) = 331 is too large for sight prime verification this iteration would be necessary: B(0-1) = 330 = 2 x 3 x 5 x 11 (these prime factors are easily verifiable) B(0 ) = 331 B(0+1) = 332 = 2 x 2 x 83 (these prime factors are easily verifiable) B(0 ) is prime, certified by B(0-1) and B(0+1) 3) The Mersenne number M =2^29-1 = 536 870 911 = A(0) is patently no prime. A(0-1) = 536 870 910 = 2 x 3 x 5 x 29 x 43 x 113 x 127 ( prime factors verifiable) A(0 ) = 536 870 911 = 233 x 1103 x 2089 A(0+1) = 536 870 912 = 2^29 ( prime factors verifiable) 4) The same Adjacent Prime factor Criterion (APC) can be used fo verify John Findley's 7-million digit Mersenne: M41=2^24,036,583 - 1 =A(0). You'll save that much time and resources for the usual independent verification. http://www.mersenne.org A(0-1)= M41 -1 A(0) =M41 A(0+1)= M41+1 5) And this rule is always true: The rightmost digit of any prime p=>11 is an element of the odd-number set nodd=[1,3,7,9] CAREFUL !! 21, 39, 57, 99 are composites, So the odd-number rule is only useful for a quick sight primality check of newly found mega-digit primes, for instance; M40=2^20,996,011-1 = ..481331395421550326484866710969127787170820477533409300972948475231983471 676 653078163294714065762855682047 http://www.mersenne.org/prime6.txt M41= 2^24,036,583 - 1 = ..49549332624134295037485542595520771846437818325642314252685868703980055603 1 269118412915067436921882733969407 http://mersenne.org/prime7.txt Now to David A. Bartizal's question: Did I miss something subtle ? Check No. 1) above . It is the main point. No. 5) is just a short-cut of your quotation All primes > 2 are odd numbers, but not all odd numbers are prime. I use the short-cut (note excluding odd number 5) to check GIMPS-Mersennes, Maybe somebody will give away a trick how to do that otherwise. Kodjo Kudiabor > > _______________________________________________ > Prime mailing list > [EMAIL PROTECTED] > http://hogranch.com/mailman/listinfo/prime _______________________________________________ Prime mailing list [EMAIL PROTECTED] http://hogranch.com/mailman/listinfo/prime
