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20378Re: [PrimeNumbers] Too big for any computer?

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  • Phil Carmody
    Jun 9, 2009
      --- On Tue, 6/9/09, Devaraj Kandadai <dkandadai@...> wrote:
      > I had  requested Ken (Kradenken)  to test whether  2^(3^53) + 99   is
      > eactly divisible by  107 or not.  He replied that
      > the number is too big for any computer.  Do you agree?

      If by "the number" you mean 2^(3^53)+99, then clearly it isn't too big for any computer as it can trivially be represented using only 9 characters. It's binary or decimal expansion is too big, but no-one's asking for its binary or decimal expansion, so that's irrelevant.

      To test whether it's divisible by 107 is trivial also, as 2^phi(107) == 1 (mod 107). So reduce 3^53 modulo phi(107):

      ? Mod(2,107)^lift(Mod(3,eulerphi(107))^53)+99
      Mod(0, 107)

      So indeed, it is divisible by 107.

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