Let F_n be the n^th Fibonacci number.

Prove: (F_mn-1) - (F_n-1)^m is divisible by (F_n)² for all m≥1 and n>1.

We should have checked that this is possibly true earlier.

The usual system of counting Fibonacci numbers is F0=0, F1=1, F2=1, F3=2, F4=3, F5=5
The problem fails for m=2, n=3:
(F5)-(F3)^2 = 8-1^2 = 7 which is not divisible by (F3)^2=2^2=4

If we count differently so that F1=0, F2=1, F3=1, F4=2, F5=3, F6=5,...,F11=55
then it fails for m=2, n=6:
(F11)-(F5)^2 = 55-3^2 = 46 which is not divisible by (F6)^2=5^2=25

Posted by Jer on 2013-05-13 13:54:05