View Single Post
 2019-05-22, 20:03 #4 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 11111100110112 Posts Statistics of Mersenne Prime Exponents Begin with the content of https://primes.utm.edu/notes/faq/NextMersenne.html We expect slightly less than 6 Mersenne primes in a 10:1 interval on exponent. M82589933 is part of an unusual run of (at least) 13 in the 107 to 108 interval. "For comparison: 1 to 10: 4 10 to 100: 6 100 to 1000: 4 1000 to 10000: 8 10000 to 100000: 6 100000 to 1000000: 5 1000000 to 10000000: 5" https://www.mersenneforum.org/showpo...70&postcount=5 (The next decade after the above, 107 to 108 is not completely double checked yet. In 108 to 109 there are relatively few tested so far, with a large amount of further computation to do.) p for Mp is 1 mod 8 at twice the frequency of 3, 5, or 7 mod 8 up to Mp50. https://www.mersenneforum.org/showpo...2&postcount=53 The Wagstaff conjecture predicts about 57 Mp below p=109. https://www.mersenneforum.org/showpo...7&postcount=55 Of the known Mersenne primes, many more are p=1 mod 4 (31) than p=3 mod 4 (19). https://www.mersenneforum.org/showpo...8&postcount=59 For #51, p= 82589933 = 1 mod 4, so 31 (60.78%) vs 19 (37.25%) 3 mod 4 and 1 (1.96%) of 2 mod 4. 31/19 ~ 1.6316. At exponents > 1000, it's ~2:1. Mersenne primes are twice as likely to be p=1 mod 4 as 3 mod 4 conjectured, with some empirical tabular support. Within p=1 (mod 4), Mersenne primes are twice as likely to have p=1 (mod 8) rather than p=5 (mod 8) https://www.mersenneforum.org/showpo...5&postcount=71 (added following 2021-08-29) There's naturally a somewhat inverse relationship between a) which p mod whatever have more found factors, and b) which p mod whatever have more found Mersenne primes. I think I may have seen it posted elsewhere, but can't find it now, that exponent p mod 24 shows considerable variation in incidence of known Mersenne primes. Code: P = 1 mod 24 8 15.69% P = 2 mod 24 1 1.96% P = 3 mod 24 1 1.96% P = 5 mod 24 9 17.65% P = 7 mod 24 8 15.69% P = 11 mod 24 3 5.88% P = 13 mod 24 3 5.88% P = 17 mod 24 11 21.57% P = 19 mod 24 5 9.80% P = 23 mod 24 2 3.92% total 51 Below p=1000, p mod 24 counts: Code: p count 1 0 2 1 3 1 5 1 7 4 11 1 13 2 17 3 19 1 23 0 Above p=1000, for known Mersenne primes, p mod 24 is observed with frequency Code: p count p n 1 8 5 8 7 4 11 2 13 1 17 8 19 4 23 2 which happens to be a table of counts consisting only of powers of two. If we break it down further to p mod 120, we get too few known primes per bin to show much. Code: p mod 120 of the 51 known Mersenne primes p count 1 3 2 1 3 1 5 1 7 3 13 1 17 4 19 1 29 1 31 1 37 1 (15 excluding 2,3,5 in p mod 120 from 0 to 40) 41 2 43 1 49 2 53 3 61 1 67 1 71 1 77 1 79 1 (13 in p mod 120 from 40 to 80) 83 1 89 3 91 2 97 3 101 3 103 3 107 2 113 2 119 1 (20 in p mod 120 from 80 to 120) Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1 Last fiddled with by kriesel on 2023-07-23 at 15:11 Reason: minor edits