Following the email by Peter-Lawrence Montgomery dated 11 Dec 2003 (quoted below).
1) There are results about some other starting values in Bas Jansen's thesis "Mersenne primes and class field theory" available at http://www.mfo.de/programme/schedule/2005/29/OWR_2005_32.pdf pages 1855-1858 (pages 57-60 in the PDF file) 2) Now we have 43 Mersenne primes known. Is it possible to get the sign of a[p-2] / 2^((p+1)/2) mod Mp for them from existing databases (without re-doing Lucas-Lehmer iterations)? Thanks, Max ========= http://www.mail-archive.com/[email protected]/msg08203.html Mersenne: Penultimate Lucas-Lehmer step Peter-Lawrence . Montgomery Thu, 11 Dec 2003 08:46:49 -0800 Let p > 2 be prime and Mp = 2^p - 1. The familiar Lucas-Lehmer test sets a[0] = 4 and a[j+1] = a[j]^2 - 2 for j >= 0. Mp is prime if and only if a[p-1] == 0 mod Mp. When Mp is prime, then a[p-2]^2 == 2 == 2*Mp + 2 = 2^(p+1) (mod Mp). Taking square roots, either a[p-2] == 2^((p+1)/2) mod Mp or a[p-2] == -2^((p+1)/2) mod Mp. Around 20 years ago I heard that nobody could predict which of these would occur. For example, p = 3 a[1] = 4 == 2^2 (mod 7) p = 5 a[3] = 194 == 2^3 (mod 31) p = 7 a[5] = 1416317954 == -2^4 (mod 127). Now that 40 Mersenne primes are known, can anyone extend this table further? That will let us test heuristics, such as whether both +- 2^((p+1)/2) seem to occur 50% of the time, and provide data to support or disprove conjectures. Peter Montgomery _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
