20090613, 19:11  #1  
Einyen
Dec 2003
Denmark
2·7·227 Posts 
Mersenne primes have highly composite p1?
http://www.mersenneforum.org/showpos...6&postcount=85
Quote:
Quote:
I took the list from 20M to 30M from the "Factoring limits" list, which is all those that has no known factor below 2^66 (20M) to 2^68 (30M). There are 587,252 primes from 20M to 30M and of them 369,166 have known factors while 218,086 are in the factoring limit list and have no known factors. I made a small program to test number of factors in p1 for all 587,252 primes and see if there was a difference: Code:
p1 factors A B  2 factors 15617=4.23% 8278=3.80% 3 factors 52822=14.31% 29439=13.50% 4 factors 81916=22.19% 47299=21.69% 5 factors 80449=21.79% 48040=22.03% 6 factors 59649=16.16% 36367=16.68% 7 factors 37016=10.03% 22935=10.51% 8 factors 20575=5.57% 12596=5.78% 9 factors 10774=2,92% 6684=3.06% 10 factors 5394=1.46% 3265=1.50% 11 factors 2571=0.70% 1686=0.77% 12 factors 1280=0.35% 784=0.36% 13 factors 612=0.17% 369=0.17% 14 factors 265=0.07% 169=0.08% 15 factors 127 92 16 factors 57 39 17 factors 28 22 18 factors 7 8 19 factors 5 5 20 factors 0 6 21 factors 1 1 22 factors 1 1 23 factors 1 1  Total 369166(100%) 218086(100%) Last fiddled with by ATH on 20090613 at 19:19 

20090613, 20:04  #2 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
I rearranged a little to show the survival rate of exponents p depending on the number of factors of p1:
Code:
p1 factors #p #survivors #suvival rate  2 factors 23895 8278 0.346 3 factors 82261 29439 0.358 4 factors 129215 47299 0.366 5 factors 128489 48040 0.374 6 factors 96016 36367 0.379 7 factors 59951 22935 0.383 8 factors 33171 12596 0.380 9 factors 17458 6684 0.383 10 factors 8659 3265 0.377 11 factors 4257 1686 0.396 12 factors 2064 784 0.380 13 factors 981 369 0.376 14 factors 434 169 0.389 My hypothesis isn't very convincing, though. By the same argument, 2^{[I]p[/I]}4 and 2^{[I]p[/I]}1 have at most the factor 3 in common, so the number of divisors in p2 (and p3 and p4 etc.) should also affect the probability that Mp is prime. Alex 
20090614, 23:04  #3  
Einyen
Dec 2003
Denmark
C6A_{16} Posts 
Quote:
Code:
p1 factors Total(20M30M) numbers without factors to 2^662^68  2 factors 49855 17960 = 36.02% 3 factors 173824 63988 = 36.81% 4 factors 218645 81127 = 37.10% 5 factors 118143 44684 = 37.83% 6 factors 25228 9692 = 38.42% 7 factors 1548 629 = 40.63% 8 factors 9 6 (=66.67%)  Total 587252 218086 If you mean all factors (not just prime factors) then the list is extensive, here is whole list (not counting 1 and p1 as factors of p1): mersennetest.txt Here is the list abbriviated by combining the factorcategories with low number of members in them: Code:
p1 factors Total(20M30M) numbers without factors to 2^662^68  2 factors 23895 8278 = 34.64% 4 factors 12264 4577 = 37.32% 6 factors 76473 27329 = 35.74% 78 factors 3355 1230 = 36.66% 10 factors 47504 17911 = 37.70% 1213 factors 985 376 = 38.17% 14 factors 98842 35686 = 36,10% 16 factors 5377 2038 = 37.90% 18 factors 10600 3991 = 37.65% 1922 factors 68947 25980 = 37.68% 2326 factors 2651 946 = 35.68% 28 factors 1955 753 = 38.52% 30 factors 66115 24718 = 37.39% 3134 factors 13651 5150 = 37.73% 3638 factors 12384 4728 = 38,18% 4046 factors 49363 18813 = 38.11% 4854 factors 3795 1435 = 37.81% 58 factors 4090 1563 = 38.22% 6162 factors 24317 9256 = 38.06% 6470 factors 12353 4747 = 38.43% 7378 factors 6840 2622 = 38.33% 7994 factors 18451 6969 = 37.77% 96106 factors 1528 577 = 37.76% 108110 factors 1234 465 = 37.68% 118 factors 2806 1091 = 38.88% 124126 factors 4652 1819 = 39.10% 128142 factors 4558 1761 = 38.64% 148158 factors 1635 634 = 38.78% 160178 factors 951 361 = 37.96% 180190 factors 2707 1076 = 39.75% 194214 factors 659 272 = 41.27% 218254 factors 1294 508 = 39.26% 258286 factors 546 230 = 42.12% 292318 factors 147 58 = 39.46% 322358 factors 141 57 = 40.43% 376382 factors 108 43 = 39.81% 394430 factors 48 28 = 58.33% 446478 factors 20 7 = 35.00% 502574 factors 11 3 = 27.27%  Total 587252 218086 (=37.14%) Last fiddled with by ATH on 20090614 at 23:11 

20090615, 13:11  #4 
Einyen
Dec 2003
Denmark
2×7×227 Posts 
Combined the categories on the last list even more (the one with all factors of p1 except 1 and p1):
Code:
p1 factors Total(20M30M) numbers without factors to 2^662^68  26 factors 112632 40184 = 35.68% 713 factors 51844 19517 = 37.65% 1418 factors 114819 41715 = 36.33% 1922 factors 68947 25980 = 37.68% 2330 factors 70721 26417 = 37.35% 3146 factors 75398 28691 = 38.05% 4862 factors 32202 12254 = 38.05% 6494 factors 37644 14338 = 38.09% 96142 factors 14778 5713 = 38.66% 148254 factors 7246 2851 = 39.35% 258382 factors 942 388 = 41.19% 394574 factors 79 38 = 48.10%  Total 587252 218086 (=37.14%) 
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