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        riddles >> medium >> A Characterization of Fibonacci Numbers
        
(Message started by: Aryabhatta on May 13^th, 2008, 4:51pm)

Title: A Characterization of Fibonacci Numbers
Post by Aryabhatta on May 13^th, 2008, 4:51pm

Show that a positive integer f is a Fibonacci number if and only if one of {5f²+4, 5f²-4} is a perfect square.

Title: Re: A Characterization of Fibonacci Numbers
Post by towr on May 14^th, 2008, 9:44am

Possibly a start, possibly not.
[hide]
g₁=1
g₂=3
g_n+2=g_n+1+g_n

odd n: g_n² = 5f_n² - 4
even n: g_n² = 5f_n² + 4
[/hide]

Title: Re: A Characterization of Fibonacci Numbers
Post by Barukh on May 14^th, 2008, 10:23am

[hide]Lucas numbers[/hide] and [hide]Pell equations[/hide]?

Title: Re: A Characterization of Fibonacci Numbers
Post by Eigenray on May 14^th, 2008, 1:23pm

Or, [hide]fundamental unit of a real quadratic number field[/hide].

Title: Re: A Characterization of Fibonacci Numbers
Post by Aryabhatta on May 14^th, 2008, 3:32pm

Yup. Well done people!