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 2020-10-10, 18:12 #12 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 10010010111112 Posts Mersenne exponents of particular forms Thread with intro, separate posts re forms The various forms or subsets are discussed here as a sort of mathematical amusement. They might also be used as an arbitrary selection method for sampling the natural number line segment for substantial black box QA testing of software algorithms. It's unlikely to be a good selection method compared to pseudorandom and boundary-informed selection methods. Mersenne rhymes As used here, a rhyme is an exponent that has in common several rightmost digits with another, in the same order. Rhymes in one number base likely will not be rhymes in most others, or only to lesser extent. Except as indicated, base ten is used here. There is no reason to believe that if an exponent p corresponds to a Mersenne prime, that any p+c*basen would also, at above the probability for a randomly chosen exponent. Yet in the category of dubious claims, people often speculate that one or more such rhymes corresponding Mersenne number are prime. For example, 82589933, 102589933. (C 2, base 10, n 7; 7digit decimal rhyme, 0x4ec38ed, 0x61d65ed 2digit hexadecimal rhyme) Empirically, it is straightforward to show that no two of the 51 known Mersenne primes' decimal exponents rhyme deeper than 3 decimal digits. A simple spreadsheet with exponents p, and cells p mod 10^n for n=1...4 and sorting by column for differing n is enough. Single digit rhymes are unavoidable in base 10 given the number of knowns. n+1-digit rhymes are a subset of n-digit rhymes. The expected number of n+1-digit rhymes is ~1/10 the number of n-digit rhymes for n>1. The number of rhymes for right digit 2 or 5 are zero, as for 4, 6, 8, or 0. Rhyming decimal exponents among the known Mersenne primes, versus number of digits 1 digit: 4 cases, 49 members of rhyme sets There are unavoidably many matches, since, after 2 and 5, there are only 4 choices for final digit, 1,3,7,9. 1 13 (31, 61, 521, 2281, 9941, 21701, 216091, 2976221, 20996011, 25964951, 42643801, 57885161, 74207281 2 (2) 3 12 (3, 13, 2203, 4253, 4423, 11213, 86243, 110503, 859433, 6972593, 24036583, 82589933) 5 (5) 7 15 (7, 17, 107, 127, 607, 3217, 19937, 44497, 1257787, 3021377, 13466917, 30402457, 32582657, 37156667, 77232917) 9 9 (19, 89, 1279, 9689, 23209, 132049, 756839, 1398269, 43112609) total 51 check 2 digit: 12 cases, 28 members 01 (21701, 42643801) 03 (3, 2203, 110503) 07 (7, 107, 607) 09 (23209, 43112609) 13 (13, 11213) 17 (17, 3217, 13466917, 77232917) 21 (521, 2976221) 33 (859433, 82589933) 57 (30402457, 32582657) 61 (61, 57885161) 81 (2281, 74207281) 89 (89, 9689) 3 digit: 2 cases, 4 members 281 (2281, 74207281) 917 (13466917, 77232917) 4 digit: 0 cases, 0 members null set 2-digit rhymes are a subset of 1-digit rhymes. 3-digit are a subset of 2-digit. 4-digit would be a subset of 3-digit. Etc. Subset of a null set is null. 5-digit and higher rhyme length are necessarily null. Straights These are exponents such as 160456789. In 2
 2020-10-12, 22:18 #13 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 111378 Posts Something about ecm memory and time scaling in prime95
 2020-10-24, 20:42 #14 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 111378 Posts GPU models no longer available to me for test GTX 480 Tesla C2075 GTX 1070

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