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F_n = F_{n-1}F_{n-2}....F_1F_0 +2, coprime fermat numbers

Source: Norwegian Mathematical Olympiad 1993 - Abel Competition p3

February 11, 2020

Fermat numbercoprimerecurrence relationnumber theory

Problem Statement

The Fermat-numbers are defined by

F_n = 2^{2^n}+1

for

n\in N

. (a) Prove that

F_n = F_{n-1}F_{n-2}....F_1F_0 +2

for

n > 0

. (b) Prove that any two different Fermat numbers are coprime