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Old 2021-10-02, 05:05   #2
Dobri
 
"刀-比-日"
May 2018

13·19 Posts
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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
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