Fibonacci of divisibility

Number Theory Level 3

let F(n) is the $n^\text{th}$ sequences of fibonacci number.

start from F(1) = 1 and F(2) = 1

is it true that $F(n)\equiv 0 \pmod4$ if and only if $n\equiv 0 \pmod6$ with every natural number n?

true false

1 solution

Stephen Mellor
Feb 15, 2018

As the Fibonacci sequence is based only on adding, we only need to consider each term mod4.

F(1) = 1
F(2) = 1
F(3) = 2
F(4) = 3
F(5) = 1 Mod 4 (5)
F(6) = 0 Mod 4 (8)
F(7) = 1 Mod 4 (13)
F(8) = 1 Mod 4 (21)

As F(1) = F(7), and F(2) = F(8), and the recurrence relation only depends on the last 2 terms, this cycle of 0 Mod 4 every 6 terms will always be true

Fibonacci of divisibility

1 solution

0 pending reports