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Explain why the number below is not 299th Fibonacci number:

222232244629420445529739893461909967206666939096499764990979600

I need an explanation

  • 8
    Because it's the $300$'th.2012-10-28
  • 1
    @RobertIsrael It really depends. If we use $0,1,1,2,\dots$ then it is the $299$th.2012-10-28
  • 5
    @RobertIsrael Ah, proof by intimidation.2012-10-28
  • 0
    One possible approach: you could try using the "closed form" expression for Fibonnaci numbers to estimate the size of the 299th and show that it's smaller than the number you give above.2012-10-28
  • 1
    Another possible approach: the Fibonacci numbers are periodic (mod $p$) for every prime p. For each $p$, you can write out a list of remainders that the Fibonacci numbers leave (mod $p$) and which indices give which remainders. Then you can divide the number by $p$ and see if the one above leaves a remainder consistent with it having an index of 299. You can try p=2, p = 3, p = 5, etc. until you find one that works.2012-10-28
  • 0
    If we start the Fibonacci sequence 1,1,2,3,5,8,13,21,34,55 ... with $F_1=1, F_2=1$ then we have that $F_r|F_{kr}$ for all positive integers $k$ - so this can be regarded as the "natural" place to start.2012-10-28
  • 0
    @Peter: Robert means that the number given in the problem is $F_{300}$ and therefore is not $F_{299}.2012-10-29

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