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Re: Prime "witness equations"

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  • rupert.wood@xtra.co.nz
    ... At the more modest levels, I have checked all primes that can have equations with at least one factor no greater than 10^8. The failing primes below 1000
    Message 1 of 19 , Jan 7, 2010
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      --- In primenumbers@yahoogroups.com, Jaroslaw Wroblewski <jaroslaw.wroblewski@...> wrote:
      >
      > 2009/12/31 rupert.wood@... <woodhodgson@...>
      >
      > > Could you please explain to the completely uninformed (me, anyway!), what
      > > these triples are and how they directly relate to the subtraction in a
      > > witness equation?
      > >
      >
      > abc triples are triples (a,b,c) of pairwise co-prime positive integers
      > satisfying a+b=c, and r<c, where r is the product of all distinct primes
      > factors of a*b*c. The measure of quality is q=log(c)/log(r), or equivalently
      > q can be defined by the equation c=r^q. Roughly speaking, we are intersted
      > in the equation a+b=c with a,b,c loaded with repeated prime factors.
      >
      > The best known triple is
      > a = 2
      > b = 3^10 * 109 = 6436341
      > c = 23^5 = 6436343
      > for which
      > r = 2 * 3 * 109 * 23 = 15042
      > and
      > q = log(c)/log(r) = 1.6299116841270481846
      >
      > It is a very reasonable conjecture to assume that there is no abc-triple
      > with a higher quality.
      >
      > A witness equation is limited to a fixed set of prime factors.
      > A witness for 523 must have a form
      > 523 = a = c-b
      > where distinct prime factors of b*c are exactly 2,3,5,7,11,13,17,19
      > Hence we have
      > r = 19# * 523
      >
      > If such a witness has quality q, then
      > c = r^q
      >
      > We can safely conjecture that there is no witness with c > r^1.63. We know
      > that any witness with r^1.4 < c < 10^20 would appear on good abc triples
      > list which contains all the existing abc triples with c < 10^20 and q > 1.4.
      >
      > Jarek
      >
      >
      > [Non-text portions of this message have been removed]
      >

      At the more modest levels, I have checked all primes that can have equations with at least one factor no greater than 10^8.

      The failing primes below 1000 are:

      397 419 433 487 541 547 557 563 599 631 647 659 701 733 743 761 769 773 797 809 823 827 839 853 859 863 877 881 907 911 937 941 947 953 967 977 983 991 997.

      The only primes above 1000 with such equations appear to be

      1303 = 8061768 - 8060465 = 2^3*3^5*11*13*29 - 5*7*17*19*23*31 and

      1601 = 11874891 - 11873290 = 3*7*17*29*31*37 - 2*5*11*13*19^2*23

      Primes above 1847 require factors greater than 10^8.
    • rupert.wood@xtra.co.nz
      I have confirmed that the list for 10^8 failures below 1000 also holds for values below 10^9.
      Message 2 of 19 , Feb 8, 2010
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        I have confirmed that the list for 10^8 "failures" below 1000 also holds for values below 10^9.

        --- In primenumbers@yahoogroups.com, "rupert.wood@..." <woodhodgson@...> wrote:
        >
        >
        >
        > --- In primenumbers@yahoogroups.com, Jaroslaw Wroblewski <jaroslaw.wroblewski@> wrote:
        > >
        > > 2009/12/31 rupert.wood@ <woodhodgson@>
        > >
        > > > Could you please explain to the completely uninformed (me, anyway!), what
        > > > these triples are and how they directly relate to the subtraction in a
        > > > witness equation?
        > > >
        > >
        > > abc triples are triples (a,b,c) of pairwise co-prime positive integers
        > > satisfying a+b=c, and r<c, where r is the product of all distinct primes
        > > factors of a*b*c. The measure of quality is q=log(c)/log(r), or equivalently
        > > q can be defined by the equation c=r^q. Roughly speaking, we are intersted
        > > in the equation a+b=c with a,b,c loaded with repeated prime factors.
        > >
        > > The best known triple is
        > > a = 2
        > > b = 3^10 * 109 = 6436341
        > > c = 23^5 = 6436343
        > > for which
        > > r = 2 * 3 * 109 * 23 = 15042
        > > and
        > > q = log(c)/log(r) = 1.6299116841270481846
        > >
        > > It is a very reasonable conjecture to assume that there is no abc-triple
        > > with a higher quality.
        > >
        > > A witness equation is limited to a fixed set of prime factors.
        > > A witness for 523 must have a form
        > > 523 = a = c-b
        > > where distinct prime factors of b*c are exactly 2,3,5,7,11,13,17,19
        > > Hence we have
        > > r = 19# * 523
        > >
        > > If such a witness has quality q, then
        > > c = r^q
        > >
        > > We can safely conjecture that there is no witness with c > r^1.63. We know
        > > that any witness with r^1.4 < c < 10^20 would appear on good abc triples
        > > list which contains all the existing abc triples with c < 10^20 and q > 1.4.
        > >
        > > Jarek
        > >
        > >
        > > [Non-text portions of this message have been removed]
        > >
        >
        > At the more modest levels, I have checked all primes that can have equations with at least one factor no greater than 10^8.
        >
        > The failing primes below 1000 are:
        >
        > 397 419 433 487 541 547 557 563 599 631 647 659 701 733 743 761 769 773 797 809 823 827 839 853 859 863 877 881 907 911 937 941 947 953 967 977 983 991 997.
        >
        > The only primes above 1000 with such equations appear to be
        >
        > 1303 = 8061768 - 8060465 = 2^3*3^5*11*13*29 - 5*7*17*19*23*31 and
        >
        > 1601 = 11874891 - 11873290 = 3*7*17*29*31*37 - 2*5*11*13*19^2*23
        >
        > Primes above 1847 require factors greater than 10^8.
        >
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