MathDB

Pythagorean Triangles and the Fibonacci Sequence

Source: Canadian Repêchage 2010: Problem 5

May 6, 2014

number theorynumber theory proposed

Problem Statement

The Fibonacci sequence is dened by

f_1=f_2=1

and

f_n=f_{n-1}+f_{n-2}

for

n\ge 3

. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that

f_{2k+1}

is the hypotenuse of a Pythagorean triangle for every positive integer

k

with

k\ge 2