MathDB
Interesting variation of Brocard's equation

Source: Bulgarian Spring Tournament 2023 12.3

March 25, 2023
factorialnumber theory

Problem Statement

Given is a polynomial ff of degree mm with integer coefficients and positive leading coefficient. A positive integer nn is good for f(x)\textit {good for f(x)} if there exists a positive integer knk_n, such that n!+1=f(n)knn!+1=f(n)^{k_n}. Prove that there exist only finitely many integers good for ff.
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