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B Prime pairs ending in _9 , _1

  1. Apr 18, 2017 #1
    45x2+15x +/-1. ... 59;61 ,209;211 ,449;451 ,779;781 ...
    45x2- 15x +/-1 ... 29;31 ,149;151 ,359;361 ,659;661 ...
    Derived from 20x2-1 can only have factors ending in the digit _1, or _9 .
     
  2. jcsd
  3. Apr 18, 2017 #2

    mfb

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    Now prove that there is an infinite set of them ;).
     
  4. Apr 18, 2017 #3
    If I use S° as a symbol to mirror x,
    The prevalence of primes in each strand ,+1 or -1 ,
    Is S° *(9/11)*(17/19)*(27/29)*(29/31)*(39/41).... does this converge to some value?
     
  5. Apr 18, 2017 #4

    mfb

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    It is strictly decreasing and bounded by 0. It will certainly converge.

    9 and 27 are not prime and 17 does not end in 1 or 9, so I'm unsure how exactly your series is defined.
     
  6. Apr 18, 2017 #5

    WWGD

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    More so if it is finite. Still, I guess this would imply there is a largest such pair. What other way is there to test for finitude of the pairs?
     
  7. Apr 18, 2017 #6

    mfb

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    We cannot conclude how many pairs there are from the convergence.
    The question how many prime pairs exist is an unsolved problem in mathematics - there are convincing arguments that their number is infinite, but there is no proof yet.
     
  8. Apr 18, 2017 #7

    WWGD

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    I know, I was going in the opposite direction: if there are finitely-many then the product converges.
     
  9. Apr 18, 2017 #8

    mfb

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    Well, it also converges with infinitely many.
     
  10. Apr 19, 2017 #9
    The series is a variant of the Euler phi function. Since the generating function (of x) contains quadratic (i.e. 2nd order)
    terms, there are 9( or 11-2) terms that are not multiples of factor_11, 17(or 19-2) terms that are not 19n multiples,
    27(or 29-2) that are not 29n multiples per 11,19,29 terms respectively.And so on.
    9,17,27 are the numerators of the fractions with the denominator 11,19,29 respectively. The product of these fractions,
    when multiplied by 11*19*29 gives 4131 (9*17*27) as the number or terms ,per 6061(11*19*29), that are relatively prime in respect to the factors 11,19,29.
     
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