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Fibonacci numbers

  1. Nov 9, 2004 #1
    Fibonacci numbers are the sequence 1,1,2,3,5,18,13,21... where after the initial two 1's, each number in the sequence is the sum of the previous two. Prove that there is no postive integer m such that the sum of every m consecutive Fibonacci numbers is odd.
    Can anyone explain to me whats the underlined part mean? Thanks alot for your help
  2. jcsd
  3. Nov 9, 2004 #2


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    It means that if you pick a natural number, m, then taking all the sets of m consecutive fibonacci numbers will always guarantee that the sum of the elements of at least one of those subsets will be even.

    For example, if m = 3 then the sets of consecutive fibonaccis would be

    {1, 1, 2}, {1, 2, 3}, {2, 3, 5}, {3, 5, 8} and so forth. Respectively, the sums of the elements of these sets are 4, 6, , 10, 16 and so forth. Obviously, when m = 3 the sums will always be even. The question is will that always be the case when m > 3.
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