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 2021-10-02, 05:05 #2 Dobri   "刀-比-日" May 2018 13×19 Posts This second post shows the minimum number of distinct exponents (repetition of same exponents is not allowed) k needed to represent a given exponent (except 2 and 3) as a sum of k smaller exponents. Note: There is no solution for the exponents 13, 521, and 756839. Here 2 + 3 + 5 + ... + 127 = 481 < 521 and 2 + 3 + 5 + ... + 216091 = 704338 < 756839. If eventually the same tendency applies to the exponent of the unknown 52nd Mersenne prime (if any), then 2 + 3 + 5 + ... + 82589933 = 580224802 < the exponent of the unknown 52nd Mersenne prime. #, k, Exponent 1, none, 2 2, none, 3 3, 2, 5 = 3 + 2 4, 2, 7 = 5 + 2 5, none, 13 6, 4, 17 = 7 + 5 + 3 + 2 7, 2, 19 = 17 + 2 8, 3, 31 = 19 + 7 + 5 9, 3, 61 = 31 + 17 + 13 10, 4, 89 = 61 + 19 + 7 + 2 11, 3, 107 = 89 + 13 + 5 12, 3, 127 = 89 + 31 + 7 13, none, 521 14, 5, 607 = 521 + 61 + 13 + 7 + 5 15, 5, 1279 = 607 + 521 + 107 + 31 + 13 16, 7, 2203 = 1279 + 521 + 127 + 107 + 89 + 61 + 19 17, 3, 2281 = 2203 + 61 + 17 18, 7, 3217 = 2281 + 521 + 127 + 107 + 89 + 61 + 31 19, 7, 4253 = 2203 + 1279 + 521 + 127 + 89 + 31 + 3 20, 4, 4423 = 4253 + 107 + 61 + 2 21, 5, 9689 = 4253 + 3217 + 2203 + 13 + 3 22, 5, 9941 = 4423 + 3217 + 2281 + 13 + 7 23, 5, 11213 = 4423 + 3217 + 2281 + 1279 + 13 24, 5, 19937 = 11213 + 4423 + 4253 + 31 + 17 25, 5, 21701 = 11213 + 9941 + 521 + 19 + 7 26, 5, 23209 = 9941 + 9689 + 2281 + 1279 + 19 27, 5, 44497 = 23209 + 11213 + 9941 + 127 + 7 28, 5, 86243 = 44497 + 21701 + 19937 + 89 + 19 29, 6, 110503 = 86243 + 19937 + 4253 + 61 + 7 + 2 30, 5, 132049 = 86243 + 44497 + 1279 + 17 + 13 31, 7, 216091 = 132049 + 44497 + 21701 + 11213 + 4423 + 2203 + 5 32, none, 756839 33, 7, 859433 = 756839 + 44497 + 23209 + 19937 + 11213 + 3217 + 521 34, 8, 1257787 = 859433 + 216091 + 132049 + 44497 + 4423 + 1279 + 13 + 2 35, 7, 1398269 = 1257787 + 86243 + 44497 + 9689 + 31 + 17 + 5 36, 9, 2976221 = 1257787 + 859433 + 756839 + 44497 + 23209 + 19937 + 11213 + 3217 + 89 37, 5, 3021377 = 2976221 + 44497 + 521 + 107 + 31 38, 9, 6972593 = 3021377 + 2976221 + 756839 + 216091 + 1279 + 521 + 127 + 107 + 31 39, 9, 13466917 = 6972593 + 3021377 + 1398269 + 1257787 + 756839 + 44497 + 11213 + 4253 + 89 40, 9, 20996011 = 13466917 + 3021377 + 2976221 + 1257787 + 132049 + 110503 + 19937 + 9941 + 1279 41, 9, 24036583 = 20996011 + 1398269 + 859433 + 756839 + 11213 + 9941 + 4253 + 607 + 17 42, 9, 25964951 = 24036583 + 859433 + 756839 + 216091 + 86243 + 4253 + 3217 + 2203 + 89 43, 8, 30402457 = 25964951 + 3021377 + 1398269 + 11213 + 4423 + 2203 + 19 + 2 44, 7, 32582657 = 24036583 + 6972593 + 1398269 + 132049 + 23209 + 19937 + 17 45, 7, 37156667 = 32582657 + 3021377 + 1398269 + 132049 + 21701 + 607 + 7 46, 9, 42643801 = 20996011 + 13466917 + 6972593 + 859433 + 216091 + 110503 + 21701 + 521 + 31 47, 7, 43112609 = 42643801 + 216091 + 132049 + 86243 + 23209 + 11213 + 3 48, 7, 57885161 = 43112609 + 13466917 + 1257787 + 44497 + 3217 + 127 + 7 49, 9, 74207281 = 25964951 + 24036583 + 20996011 + 2976221 + 216091 + 9689 + 4253 + 2203 + 1279 50, 7, 77232917 = 74207281 + 1398269 + 859433 + 756839 + 9689 + 1279 + 127 51, 7, 82589933 = 32582657 + 25964951 + 20996011 + 3021377 + 21701 + 3217 + 19 Last fiddled with by Dobri on 2021-10-02 at 06:29