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p!|(a +1) if $p!|(a^p + 1)

Source: China Northern MO 2012 p8 CNMO

May 4, 2024
number theorydivides

Problem Statement

Assume pp is a prime number. If there is a positive integer aa such that p!∣(ap+1)p!|(a^p + 1), prove that :
(1) (a+1,ap+1a+1)=p(a+1, \frac{a^p+1}{a+1}) = p
(2) ap+1a+1\frac{a^p+1}{a+1} has no prime factors less than pp.
(3) p!∣(a+1)p!|(a +1) .
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