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Fibonacci sequence=>∀m; ∃ k: m|(a_k^4-a_k-2)

Source: Czech-Polish-Slovak Match 2007-P2

September 14, 2011

number theory proposednumber theory

Problem Statement

The Fibonacci sequence is deﬁned by

a_1=a_2=1

and

a_{k+2}=a_{k+1}+a_k

for

k\in\mathbb N.

Prove that for any natural number

m,

there exists an index

k

such that

a_k^4-a_k-2

is divisible by

m.

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