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.

Praneeth, do you have a solution to this?  The result in the paper you mention here is incorrect.  Result (8) is correct but Observation (1) following is not.

For example, taking p=31 and A=5, S'(A) = {2, 4, 8, 16}, with sum 30, not 61 as predicted by the result.  More generally, if p=2^A - 1 is a Mersenne prime, S(p, A) = p, not (A-1)/2*p.

Nevertheless, it is an interesting problem.  I would guess that any sort of closed form would be a deep result.

Perhaps there is an elementary solution to the following: find all solutions to S(p, A) = p.

Posted by Eigenray on 2007-08-31 17:03:47