View Single Post
Old 2021-10-02, 03:36   #1
Dobri
 
"Καλός"
May 2018

32310 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