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Prove these two problems on Fibonacci sequence.

Source: MTRP 2018 Class 11-Short Answer Type Question: Problem 5 :-

February 17, 2021

Fibonacci sequenceremainderPeriodic sequencepolynomialalgebra

Problem Statement

[*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by

n

is periodic, where

n

is a natural number. [*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number

n,

P(n)

is a prime.