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q^{(q+1)^n}+1 is divisible by (q + 1)^{n+1} but not (q + 1)^{n+2}

Source: Polish MO Recond Round 1982 p5

September 9, 2024

number theorydividesdivisible

Problem Statement

Let

q

be an even positive number. Prove that for every natural number

n

number

q^{(q+1)^n}+1

is divisible by

(q + 1)^{n+1}

but not divisible by

(q + 1)^{n+2}