mersenneforum.org Additive Properties of the Exponents of Known Mersenne Primes
 Register FAQ Search Today's Posts Mark Forums Read

 2021-10-02, 03:36 #1 Dobri   "刀-比-日" May 2018 2×7×17 Posts Additive Properties of the Exponents of Known Mersenne Primes This thread is intended to provide a collection of empirical observations concerning the additive properties of the exponents of known Mersenne primes. This initial post shows the minimum number of exponents (repetition of same exponents is allowed) k needed to represent a given exponent (except 2 and 3) as a sum of k smaller exponents. For the known Mersenne primes, the value of k does not exceed 9. Note: A related branch of number theory is called additive number theory, see https://en.wikipedia.org/wiki/Additive_number_theory. #, k, Exponent 1, none, 2 2, none, 3 3, 2, 5 = 3 + 2 4, 2, 7 = 5 + 2 5, 3, 13 = 5 + 5 + 3 6, 3, 17 = 7 + 5 + 5 7, 2, 19 = 17 + 2 8, 3, 31 = 13 + 13 + 5 9, 3, 61 = 31 + 17 + 13 10, 4, 89 = 61 + 13 + 13 + 2 11, 3, 107 = 89 + 13 + 5 12, 3, 127 = 61 + 61 + 5 13, 5, 521 = 127 + 127 + 89 + 89 + 89 14, 5, 607 = 521 + 31 + 19 + 19 + 17 15, 5, 1279 = 521 + 521 + 127 + 107 + 3 16, 6, 2203 = 607 + 521 + 521 + 521 + 31 + 2 17, 3, 2281 = 2203 + 61 + 17 18, 5, 3217 = 1279 + 1279 + 521 + 107 + 31 19, 7, 4253 = 1279 + 1279 + 521 + 521 + 521 + 127 + 5 20, 3, 4423 = 2203 + 2203 + 17 21, 5, 9689 = 3217 + 3217 + 3217 + 19 + 19 22, 5, 9941 = 4253 + 2203 + 2203 + 1279 + 3 23, 5, 11213 = 4253 + 3217 + 3217 + 521 + 5 24, 5, 19937 = 9689 + 9689 + 521 + 19 + 19 25, 5, 21701 = 9689 + 9689 + 2203 + 89 + 31 26, 5, 23209 = 9689 + 9689 + 3217 + 607 + 7 27, 5, 44497 = 19937 + 19937 + 2281 + 2281 + 61 28, 5, 86243 = 44497 + 21701 + 19937 + 89 + 19 29, 5, 110503 = 44497 + 44497 + 11213 + 9689 + 607 30, 5, 132049 = 86243 + 44497 + 1279 + 17 + 13 31, 7, 216091 = 86243 + 86243 + 21701 + 21701 + 107 + 89 + 7 32, 8, 756839 = 216091 + 216091 + 216091 + 86243 + 21701 + 607 + 13 + 2 33, 7, 859433 = 756839 + 44497 + 21701 + 21701 + 11213 + 2203 + 1279 34, 7, 1257787 = 859433 + 132049 + 132049 + 132049 + 2203 + 2 + 2 35, 7, 1398269 = 1257787 + 86243 + 44497 + 9689 + 19 + 17 + 17 36, 7, 2976221 = 1398269 + 1398269 + 132049 + 23209 + 21701 + 2203 + 521 37, 5, 3021377 = 2976221 + 44497 + 521 + 107 + 31 38, 7, 6972593 = 2976221 + 2976221 + 756839 + 216091 + 44497 + 2203 + 521 39, 9, 13466917 = 2976221 + 2976221 + 2976221 + 2976221 + 1398269 + 132049 + 23209 + 4253 + 4253 40, 9, 20996011 = 6972593 + 6972593 + 3021377 + 1257787 + 1257787 + 756839 + 756839 + 107 + 89 41, 7, 24036583 = 20996011 + 3021377 + 9689 + 4423 + 2281 + 2281 + 521 42, 7, 25964951 = 24036583 + 1257787 + 216091 + 216091 + 216091 + 21701 + 607 43, 8, 30402457 = 13466917 + 6972593 + 6972593 + 2976221 + 9689 + 4423 + 19 + 2 44, 7, 32582657 = 24036583 + 6972593 + 1398269 + 132049 + 23209 + 19937 + 17 45, 7, 37156667 = 30402457 + 2976221 + 1257787 + 859433 + 859433 + 756839 + 44497 46, 7, 42643801 = 32582657 + 6972593 + 2976221 + 44497 + 44497 + 23209 + 127 47, 7, 43112609 = 37156667 + 2976221 + 2976221 + 2203 + 1279 + 13 + 5 48, 7, 57885161 = 25964951 + 25964951 + 2976221 + 2976221 + 2203 + 607 + 7 49, 8, 74207281 = 37156667 + 24036583 + 6972593 + 2976221 + 2976221 + 44497 + 44497 + 2 50, 5, 77232917 = 74207281 + 3021377 + 3217 + 521 + 521 51, 7, 82589933 = 32582657 + 25964951 + 20996011 + 3021377 + 21701 + 3217+ 19 Last fiddled with by Dobri on 2021-10-02 at 05:54
 2021-10-02, 05:05 #2 Dobri   "刀-比-日" May 2018 2×7×17 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
 2021-10-05, 04:22 #3 Dobri   "刀-比-日" May 2018 2×7×17 Posts Assuming that the prime exponent of the next unknown 52nd Mersenne prime (if any) can be represented as a sum of k smaller known prime exponents, the number of prime exponents to be tested is reduced roughly by an order of magnitude as compared to the total number of prime exponents. Eliminating the prime exponents that have already been verified/factored, said number is reduced roughly by two orders of magnitude. k, Number of Prime Exponents (repetition of same exponents is allowed in the summation of k smaller exponents), Number of Remaining Untested/Unverified Prime Exponents 2, 5, 0 3, 3162, 203 4, 2801, 193 5, 306293, … 6, 303972, … 7, 8674167, … 8, 8685411, … k, Number of Prime Exponents (repetition of same exponents is not allowed in the summation of k smaller exponents), Number of Remaining Untested/Unverified Prime Exponents 2, 5, 0 3, 2952, 179 4, 2412, … 5, 224493, … 6, 214739, … 7, 5541799, … 8, 5500506, … Note: The ellipsis indicates that it is preferable not to congest the server in checking the status (Untested or Unverified) of so many prime exponents. One could check specific narrow ranges instead.
 2021-10-05, 06:56 #4 Dobri   "刀-比-日" May 2018 2×7×17 Posts If the prime exponent of the next unknown 52nd Mersenne prime (if any) cannot be represented as a sum of k < 10 smaller known prime exponents and it is not greater than the total sum of all smaller known prime exponents, then it will be a prime exponent of a Mersenne prime that breaks with the current observations of the limited sample size, truly a first of its kind. Last fiddled with by Dobri on 2021-10-05 at 06:56

 Similar Threads Thread Thread Starter Forum Replies Last Post drkirkby Miscellaneous Math 39 2021-08-24 21:08 sweety439 sweety439 0 2021-06-24 02:29 ProximaCentauri Miscellaneous Math 15 2014-12-25 14:26 Lee Yiyuan Miscellaneous Math 60 2011-03-01 12:22 kurtulmehtap Math 31 2011-01-10 00:15

All times are UTC. The time now is 06:38.

Tue Dec 7 06:38:38 UTC 2021 up 137 days, 1:07, 0 users, load averages: 1.00, 1.15, 1.28