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Re: Square factors of b^p-1

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  • djbroadhurst
    ... Also 4232737^17591459 = 1 mod 35182919^2 15865919^17591459 = 1 mod 35182919^2 4144001^19276511 = 1 mod 38553023^2 13038863^19276511 = 1 mod 38553023^2
    Message 1 of 81 , Oct 3, 2011
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      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      >> Puzzle 3a: Find primes b1 < p1 < q, b2 < p2 < q, b1 < b2, s.t.
      >> b1^p1 = b2^p2 = 1 mod q^2, (with no limits on q^2).
      >
      > At q^2 > 10^15, I found
      >
      > 1634447^18090263 = 1 mod 36180527^2
      > 1773371^18090263 = 1 mod 36180527^2
      >
      > 3633673^26867849 = 1 mod 53735699^2
      > 21160553^26867849 = 1 mod 53735699^2

      Also

      4232737^17591459 = 1 mod 35182919^2
      15865919^17591459 = 1 mod 35182919^2

      4144001^19276511 = 1 mod 38553023^2
      13038863^19276511 = 1 mod 38553023^2

      2963911^27536213 = 1 mod 55072427^2
      3658729^27536213 = 1 mod 55072427^2

      David
    • Walter Nissen
      Hi , all , Not even draft yet , let alone published : http://upforthecount.com/math/nnp.html still needs a lot of work . But , I have updated the top of the
      Message 81 of 81 , Nov 29, 2011
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        Hi , all ,

        Not even draft yet , let alone published :
        http://upforthecount.com/math/nnp.html
        still needs a lot of work .
        But , I have updated the top of the main table :
        http://upforthecount.com/math/nnp2.txt
        with complete factorizations now extending through
        np ( 90 ) .

        When is np ( n ) prime ?
        So far , only for 1 , 2 and 3 .
        When n mod 6 = 4 , np ( n ) has at least 3 algebraic
        factors , 3 and n^2 + n + 1 ( twice ) .
        But what about the residue ( primitive part ? ) ?
        When is this prime ?
        n t n^n + (n+1)^(n+1)
        4 4 3 * 7 * 7 * 23
        16 6 3 * 7 * 7 * 13 * 13 * 34041259347101651
        Any others ?

        Cheers ,

        Walter
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