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Exponent of primes

Source: Indian IMOTC 2013, Practice Test 1, Problem 1

May 6, 2013
number theory proposednumber theory

Problem Statement

For a prime pp, a natural number nn and an integer aa, we let Sn(a,p)S_n(a,p) denote the exponent of pp in the prime factorisation of apn−1a^{p^n} - 1. For example, S1(4,3)=2S_1(4,3) = 2 and S2(6,2)=0S_2(6,2) = 0. Find all pairs (n,p)(n,p) such that Sn(2013,p)=100S_n(2013,p) = 100.
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