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Fun primes

Source: Mexico National Olympiad Mock Exam 2021 P6

November 12, 2021
number theory

Problem Statement

Let aa and bb be fixed positive integers. We say that a prime pp is fun if there exists a positive integer nn satisfying the following conditions:
[*]pp divides an!+ba^{n!} + b. [*]pp divides a(n+1)!+ba^{(n + 1)!} + b. [*]p<2n2+1p < 2n^2 + 1.
Show that there are finitely many fun primes.
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