MathDB

5

Part of 2010 Canadian Mathematical Olympiad Qualification Repechage

Problems(1)

Pythagorean Triangles and the Fibonacci Sequence

Source: Canadian Repêchage 2010: Problem 5

5/6/2014
The Fibonacci sequence is de ned by f1=f2=1f_1=f_2=1 and fn=fn−1+fn−2f_n=f_{n-1}+f_{n-2} for n≥3n\ge 3. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that f2k+1f_{2k+1} is the hypotenuse of a Pythagorean triangle for every positive integer kk with k≥2k\ge 2
number theorynumber theory proposed