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2017 Algebra/NT #9: Fibonacci number divisible by 127

Source:

February 19, 2017

modular arithmeticFibonacci sequencenumber theory

Problem Statement

The Fibonacci sequence is defined as follows:

F_0=0

F_1=1

, and

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

for all integers

n\ge 2

. Find the smallest positive integer

m

such that

F_m\equiv 0 \pmod {127}

and

F_{m+1}\equiv 1\pmod {127}