Additive Properties of the Exponents of Known Mersenne Primes

[Dobri]

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

[Dobri]

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 52^nd 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

[Dobri]

Assuming that the prime exponent of the next unknown 52^nd 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]

If the prime exponent of the next unknown 52^nd 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.