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no (a_s-a_t)/(s-t) is an integer - if p is prime divisor of s-t, then p/(a-1)

Source: Balkan BMO Shortlist 2015 N3

August 5, 2019
prime divisorIntegernumber theorydividesdivisor

Problem Statement

Let aa be a positive integer. For all positive integer n, we define an=1+a+a2+…+an−1. a_n=1+a+a^2+\ldots+a^{n-1}. Let s,ts,t be two different positive integers with the following property: If pp is prime divisor of s−ts-t, then pp divides a−1a-1. Prove that number as−ats−t\frac{a_{s}-a_{t}}{s-t} is an integer.
(FYROM)
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