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f(n) = g^n + 1 where g is even positive integer

Source: 1991 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p2

November 20, 2022
number theorydivides

Problem Statement

Let gg be an even positive integer and f(n)=gn+1f(n) = g^n + 1 , (nN)(n \in N^* ). Prove that for every positive integer nn we have: a) f(n)f(n) divides each of the numbers f(3n),f(5n),f(7n)f(3n), f(5n), f(7n) b) f(n)f(n) is relative prime to each of the numbers f(2n),f(4n),f(6n),...f(2n), f(4n),f(6n),...