Let F
_{n} be the n
^{th} Fibonacci number.
Prove: (F_{mn1})  (F_{n1})^{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 = 81^2 = 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 = 553^2 = 46 which is not divisible by (F6)^2=5^2=25

Posted by Jer
on 20130513 13:54:05 