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RE Infinite Number of Twin Primes

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  • Jose Ramón Brox
    You have not proved a thing; you merely launched a new hypothesis. You are not proving anywhere that, for a fixed factorial, it exists a twin prime that added
    Message 1 of 2 , May 5, 2005
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      You have not proved a thing; you merely launched a new hypothesis.

      You are not proving anywhere that, for a fixed factorial, it exists a twin prime that
      added to the factorial will always give another twin pair. You only give some evidence.
      Indeed, a very very little evidence for it. Moreover, you don't need to prove that the sum
      is a twin, if you can simply prove that it exists a twin for every factorial, then you
      would have proven the TP conjecture! But you can't prove it with your idea, unless you
      build a chain of numbers that are always twins (you have to prove it) and you prove that
      adding that twins to factorials give new twins. That's it, to make explicit the twins that
      your chain will use.

      Nevertheless, your idea about that relation between two twin primes seems beautiful to me.
      Will all the twin primes be related to a factorial and another twin prime? If it is true,
      is it trivial? If it is not trivial, is there a simple formula to regulate this fact? How
      many primes will you add to the same factorial before you change to the next factorial?
      Because factorials grow a lot faster than twin primes do (at least I think so)

      Or will it be a consecuence of the small law of numbers? Can you go further with your
      checking? Go up with the twins to, say, 10^8, if you can, and then return with the results
      if you think they are relevant.

      Jose Brox

      ----- Original Message -----
      From: "Milton Brown" <miltbrown@...>
      To: "primenumbers" <primenumbers@yahoogroups.com>
      Sent: Thursday, May 05, 2005 10:54 PM
      Subject: [PrimeNumbers] Infinite Number of Twin Primes


      Here is a demonstration that there are an infinite number of Twin Primes,
      and a method of finding them.

      Consider the following twin primes:

      (5,7)
      (11,13) (17, 19)
      (29,31) (41,43)
      (149,151) (179,181) ...

      Now put the appropriate factorial ahead of them

      (2!+3,2!+5)
      (3!+5,3!+7) (3!+11,3!+13) note (3!+17,3!+19=25) does not woork
      (4!+5,4!+7) (4!+17,4!+19)
      (5!+29,5!+31) (5!+59,5!+61)

      This is the pattern: Start at the next factorial and add prior pairs of twin primes,
      until you obtain a pair of twin primes for that factorial.

      The equivalent theorem is "There is always a pair of twin primes between a factorial
      and its successor factorial."

      Then Ininite Twin Primes are proved since 2!, 3!, ..., n!, ... are infinite in number.

      The familiar formula:

      n!+2, n!+3, ..., n!+n

      shows that these numbers are all composite,
      but for this value of n we start testing the previous twin primes at

      n!+(n+1) and (n-1)! > n+1 for n>4. This is easily proved by induction on n:

      (n-1)! > n+1
      n! = n(n-1)! > n^2+n > n+2 since

      n^2 > 2.

      Milton L. Brown

      [Non-text portions of this message have been removed]






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    • Jose Ramón Brox
      I found that 4!+857, 4!+859 adds (881,883) allowing the primes generated by the n factorial to go backwards to be checked with former factorials. But then I
      Message 2 of 2 , May 5, 2005
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        I found that

        4!+857, 4!+859 adds (881,883)

        allowing the primes generated by the n factorial to go "backwards" to be checked with
        former factorials.

        But then I got stuck. No twin primes with (881,883) with n! from 3 to 7.

        Jose Brox

        PS Milton still has an escape, if he has a way to determine a twin that added with the
        factorial will give another twin... but this is silly, cause that method would prove TP
        conjecture itself!



        ----- Original Message -----
        From: "Décio Luiz Gazzoni Filho" <decio@...>
        To: <primenumbers@yahoogroups.com>
        Sent: Friday, May 06, 2005 1:13 AM
        Subject: Re: [PrimeNumbers] Infinite Number of Twin Primes


        On Thursday 05 May 2005 19:49, you wrote:
        > At 04:54 PM 5/5/2005, Milton Brown wrote:
        > >This is the pattern: Start at the next factorial and add prior pairs of
        > >twin primes,
        > >until you obtain a pair of twin primes for that factorial.
        >
        > How do you know that will work (that you will always obtain a pair of twins
        > for that factorial)?

        The answer is, he doesn't, because it won't work. This one didn't even need a
        computer search; it fell to a search by hand. I would appreciate a third
        check (I've already double-checked locally).

        Start from (3,5).

        2!+3, 2!+5 adds (5,7). Values found until now: (3,5), (5,7).

        3!+5, 3!+7 adds (11,13). 3!+11, 3!+13 adds (17,19).
        Values found until now: (3,5), (5,7), (11,13), (17,19).

        4!+5, 4!+7 adds (29,31). 4!+17, 4!+19 adds (41,43).
        Values found until now: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43).

        5!+17, 5!+19 adds (137,139). 5!+29, 5!+31 adds (149,151). 5!+149, 5!+151 adds
        (269,271).
        Values found until now: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43),
        (137,139), (149,151), (269,271).

        6!+137, 6!+139 adds (857,859).
        Values found until now: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43),
        (137,139), (149,151), (269,271), (857,859).

        Finally, 7! doesn't produce any values. And thus Milton's conjecture is
        demolished, as about anything that he brainfarts on this list.

        Décio


        [Non-text portions of this message have been removed]




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