wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> Triangle and Fib
        
(Message started by: Christine on Mar 5^th, 2013, 3:24pm)

Title: Triangle and Fib
Post by Christine on Mar 5^th, 2013, 3:24pm

Can you explain why there are no triangles of any kind with sides the fibonacci numbers (being different)?

Title: Re: Triangle and Fib
Post by peoplepower on Mar 5^th, 2013, 4:41pm

The problem is that [hide]your longest side will necessarily be too long for the triangle to not degenerate.

Say you have side lengths F_k,F_m,F_n with k<m<n. Then F_k+F_m <= F_m-1+F_m <= F_n.[/hide]

Title: Re: Triangle and Fib
Post by Christine on Mar 11^th, 2013, 1:18pm

I see. Thanks

Title: Re: Triangle and Fib
Post by Christine on Apr 4^th, 2013, 12:22pm

Can you find all integer solutions to

a^2 + b^2 + c^2 + d^2 = e^2
where a, b, c, d, e are fibonacci numbers

a^2 + b^2 + c^2 + d^2 + e^2 = f^2
where a, b, c, d, e, f are fibonacci numbers

Title: Re: Triangle and Fib
Post by towr on Apr 4^th, 2013, 1:01pm

Do they need to be different and/or non-zero? (Just to rule out the trivial case where you pick all but one on the left-hand side 0.)

Title: Re: Triangle and Fib
Post by Christine on Apr 4^th, 2013, 1:19pm

on 04/04/13 at 13:01:52, towr wrote:

Do they need to be different and/or non-zero? (Just to rule out the trivial case where you pick all but one on the left-hand side 0.)

Let's ignore the trivial cases.

Title: Re: Triangle and Fib
Post by pex on Apr 4^th, 2013, 1:34pm

For the sum of four squares, there's of course
1² + 1² + 1² + 1² = 2².

For sums of five squares, small solutions are
2² + 2² + 2² + 2² + 3² = 5², and
1² + 2² + 3² + 5² + 5² = 8².

There are no other solutions with all numbers positive within the first 38 Fibonacci numbers. (I don't really feel like searching for something with high enough precision to check higher numbers.) It might well be that there are no further solutions... anyone?