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Fix a positive integer

a > 1

. Consider triples

(f(x), g(x), h(x))

of polynomials with integer coefficients, such that 1.

f

is a monic polynomial with

\deg f \ge 1

. 2. There exists a positive integer

N

such that

g(x)>0

for

x \ge N

and for all positive integers

n \ge N

, we have

f(n) \mid a^{g(n)} + h(n)

.
Find all such possible triples.
Proposed by Mainak Ghosh and Rijul Saini

Problem Statement