Let S(A) be {1,2,3,...,(A-1)} and S'(A,B) be the set of elements X from S(A) that satisfy X^B mod A=1. Assuming that A is prime, find the sum of all the elements of S'(A,B) in terms of A and B.
(In reply to
see a pattern here? by Charlie)
For these prime values of A, the sums appear to repeat in a cycle of A-1 entries as B is increased. In the table below, two cycles are shown for each A. Values for various B's may extend over more than one line; those for B=17, 33, etc. appear under the A column because of the way the program is written; the B values increase by 16 each line within a given A value.
B 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
34 35 36 37 38 39 40 41 42 43 44 45 46
A
2 1 1
3 3 1 3 1
5 5 1 10 1 5 1 10 1
7 7 7 7 1 21 1 7 7 7 1 21 1
11 11 1 11 22 11 1 11 1 55 1 11 1 11 22 11
1 11 1 55 1
13 13 13 26 1 39 1 26 13 13 1 78 1 13 13 26
1 39 1 26 13 13 1 78 1
17 17 1 34 1 17 1 68 1 17 1 34 1 17 1 136
1 17 1 34 1 17 1 68 1 17 1 34 1 17 1 136
1
19 19 19 19 1 57 1 19 76 19 1 57 1 19 19 19
1 171 1 19 19 19 1 57 1 19 76 19 1 57 1 19
19 19 1 171 1
23 23 1 23 1 23 1 23 1 23 92 23 1 23 1 23
1 23 1 23 1 253 1 23 1 23 1 23 1 23 1 23
92 23 1 23 1 23 1 23 1 23 1 253 1
29 29 1 58 1 29 116 58 1 29 1 58 1 203 1 58
1 29 1 58 116 29 1 58 1 29 1 406 1 29 1 58
1 29 116 58 1 29 1 58 1 203 1 58 1 29 1 58
116 29 1 58 1 29 1 406 1
31 31 31 31 31 93 1 31 31 155 1 93 1 31 186 31
1 93 1 155 31 31 1 93 31 31 31 31 1 465 1 31
31 31 31 93 1 31 31 155 1 93 1 31 186 31 1 93
1 155 31 31 1 93 31 31 31 31 1 465 1
37 37 37 74 1 111 1 74 148 37 1 222 1 37 37 74
1 333 1 74 37 37 1 222 1 37 148 74 1 111 1 74
37 37 1 666 1 37 37 74 1 111 1 74 148 37 1 222
1 37 37 74 1 333 1 74 37 37 1 222 1 37 148 74
1 111 1 74 37 37 1 666 1
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Posted by Charlie
on 2007-07-20 23:16:21 |
One pattern
