mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Blogorrhea > Dobri

Reply
 
Thread Tools
Old 2021-10-02, 03:36   #1
Dobri
 
"刀-比-日"
May 2018

2·7·17 Posts
Post 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
Dobri is offline   Reply With Quote
Old 2021-10-02, 05:05   #2
Dobri
 
"刀-比-日"
May 2018

2·7·17 Posts
Default

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
Dobri is offline   Reply With Quote
Old 2021-10-05, 04:22   #3
Dobri
 
"刀-比-日"
May 2018

2·7·17 Posts
Default

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.
Dobri is offline   Reply With Quote
Old 2021-10-05, 06:56   #4
Dobri
 
"刀-比-日"
May 2018

2×7×17 Posts
Default

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
Dobri is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
More twin primes below Mersenne exponents than above Mersenne exponents. drkirkby Miscellaneous Math 39 2021-08-24 21:08
Interesting properties about Mersenne(-related) exponents and Wagstaff(-related) exponents sweety439 sweety439 0 2021-06-24 02:29
Sophie-Germain primes as Mersenne exponents ProximaCentauri Miscellaneous Math 15 2014-12-25 14:26
Assorted formulas for exponents of Mersenne primes Lee Yiyuan Miscellaneous Math 60 2011-03-01 12:22
Properties of Mersenne numbers kurtulmehtap Math 31 2011-01-10 00:15

All times are UTC. The time now is 04:12.


Wed Dec 1 04:12:48 UTC 2021 up 130 days, 22:41, 1 user, load averages: 1.36, 1.43, 1.37

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.