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Quotient of partial sums in geometric series is an integer

Source: Macedonian TST for IMO 2013 - P3 day 2

March 28, 2021

geometric seriesnumber theory

Problem Statement

Let

a

and

n>0

be integers. Define

a_{n} = 1+a+a^2...+a^{n-1}

. Show that if

p|a^p-1

for all prime divisors of

n_{2}-n_{1}

, then the number

\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}

is an integer.