MathDB

(Not) Almost Fibonacci

Source: Brazilian Undergrad MO 2020 Problem 2

May 17, 2022

Brazilian Undergrad MO 2020number theorySequencesBrazilian Undergrad MO

Problem Statement

For a positive integer

a

, define

F_1 ^{(a)}=1

F_2 ^{(a)}=a

and for

n>2

F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}

. A positive integer is fibonatic when it is equal to

F_n ^{(a)}

for a positive integer

a

and

n>3

. Prove that there are infintely many not fibonatic integers.