Let A be an integer, P an odd prime and n=3 be the smallest integer for which A^n - 1 is divisible by P.

Determine the smallest integer m for which (A+1)^m - 1 is divisible by P.

(In reply to

some thoughts by Dennis)

I agree with you that M=6 is the answer. P divides (A©÷+2A+1) and (A+1)^6-1 = A(A+2)(A©÷+3A+3)(A©÷+A+1)

I have also showed to myself that (A+1)^m-1, where 1¡Âm¡Ã5, does not possess the factor (A©÷+2A+1), but since (A©÷+2A+1) is not necessarily prime, I am unable to show that (A+1)^m-1, where 1¡Âm¡Ã5, never posess the factor p.

Are you able to show that m=6 is the smallest integer even if A is really big?