On Wed, 10 Jul 2002, Mersenne Digest wrote: > From: "Frank_A_L_I_N_Y" <[EMAIL PROTECTED]> > Subject: Mersenne: m(127) by machine > > Does anyone know which parts of the ll test are due to lucas and which = > to Lehmer? > Specifically was the MODing at the end of the function known to Lucas, = > or was it an addition of Lehmer's. I am sure someone else can answer this better than I can, but: For a start, you may want to peek at the May _Mathematics Magazine_ which has an article "A brief history of factoring and primality testing B.C. (before computers) by R A Mollin. Here's a passage from that article: To see how Lucas determined M127 is prime we state the following result that was known to Lucas... Suppose that F_k is the kth Fibonacci number and n is given. If n = +- 3 mod 10 and n divides F_n+1, but n does not divide F_m for all divisors of m with 1<=m<=n, then n is prime. Also if n = +=1 mod 10 and n divides F_n-1 but n does not divide F_m for all divisors m of n with 1<=m<=n-2 then n is prime. Based on this result, all Lucas had to establish was that M127 divides F_2^127 but M127 does not divide F_2^m for all m<127. He did this in 1876, using methods that led to a primality test... [they quote the Lucas-Lehmer test we all know and love]... Lucas knew only that teh test was sufficient for primality, and this only for certain restricted types of values of n. In 1930, Lehmer proved both that the condition is necessary and that the test holds for any value of n. [end quote] As I read that passage, Lucas came up with the testing method in its entirety, and knew that if a number passed the LL test it was prime, but didn't use it the way we do because failing the LL test wasn't known to be proof of compositeness before Lehmer. GRB _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
