Loading ...
Sorry, an error occurred while loading the content.

Infinite Number of Twin Primes

Expand Messages
  • Milton Brown
    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,
    Message 1 of 13 , May 5, 2005
    View Source
    • 0 Attachment
      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]
    • Jud McCranie
      ... How do you know that will work (that you will always obtain a pair of twins for that factorial)?
      Message 2 of 13 , May 5, 2005
      View Source
      • 0 Attachment
        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)?
      • Décio Luiz Gazzoni Filho
        ... 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
        Message 3 of 13 , May 5, 2005
        View Source
        • 0 Attachment
          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]
        • Jud McCranie
          ... I knew that, of course, and was pressing him.
          Message 4 of 13 , May 5, 2005
          View Source
          • 0 Attachment
            At 07:13 PM 5/5/2005, Décio Luiz Gazzoni Filho wrote:

            > 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,

            I knew that, of course, and was pressing him.
          • Milton Brown
            Perhaps a computer search would be better for you. Or, see the link http://primes.utm.edu/lists/small/1ktwins.txt (7!+59, 7!+61) = (5099, 5101) is a prime
            Message 5 of 13 , May 5, 2005
            View Source
            • 0 Attachment
              Perhaps a computer search would be better for you.

              Or, see the link http://primes.utm.edu/lists/small/1ktwins.txt

              (7!+59, 7!+61) = (5099, 5101) is a prime pair.

              Milton L. Brown


              > [Original Message]
              > From: D�cio Luiz Gazzoni Filho <decio@...>
              > To: <primenumbers@yahoogroups.com>
              > Date: 5/5/2005 4:14:35 PM
              > 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.
              >
              > D�cio
              >
              >
              > [Non-text portions of this message have been removed]
              >
              >
              >
              >
              > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
              > The Prime Pages : http://www.primepages.org/
              >
              >
              > Yahoo! Groups Links
              >
              >
              >
              >
              >
            • Décio Luiz Gazzoni Filho
              ... Milton, are you really this stupid, or do you somehow enjoy being humiliated in public? In the latter case, you should seek psychiatric help. ... Emphasis
              Message 6 of 13 , May 5, 2005
              View Source
              • 0 Attachment
                On Thursday 05 May 2005 21:36, Milton Brown wrote:
                > Perhaps a computer search would be better for you.
                >
                > Or, see the link http://primes.utm.edu/lists/small/1ktwins.txt
                >
                > (7!+59, 7!+61) = (5099, 5101) is a prime pair.

                Milton, are you really this stupid, or do you somehow enjoy being humiliated
                in public? In the latter case, you should seek psychiatric help.

                From your message that started the thread:

                > 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.

                Emphasis on `prior pairs'. Now tell me which values of n,k generate (n! + k,
                n! + k + 2) = (59,61).

                Décio


                [Non-text portions of this message have been removed]
              • Décio Luiz Gazzoni Filho
                ... Moreover, a PARI/GP search plus a clever shell script using PFGW (guess I should learn PFGW s scripting language instead, but...) confirms my previous
                Message 7 of 13 , May 5, 2005
                View Source
                • 0 Attachment
                  On Thursday 05 May 2005 20:13, you wrote:
                  > Finally, 7! doesn't produce any values. And thus Milton's conjecture is
                  > demolished, as about anything that he brainfarts on this list.

                  Moreover, a PARI/GP search plus a clever shell script using PFGW (guess I
                  should learn PFGW's scripting language instead, but...) confirms my previous
                  findings, and adds failures at n! for n = 10, 12-14, 16, 17, 22-64, 66-70,
                  72-135, 137-361, 363-683,685-900; that's as far as I'm willing to take this
                  search. I have reason to believe that only finitely many twin prime pairs are
                  generated by this process.

                  Additionally, in the course of running this code, I discovered a bug in
                  PARI/GP 2.2.9:

                  ? isprime(42542905343533366778773944705953203289361426945380795183689)
                  *** isprime: impossible inverse modulo: Mod(0,
                  42542905343533366778773944705953203289361426945380795183689).

                  Ironically, Milton's work was not 100% in vain.

                  Décio

                  PS: in principle one can actually check my conjecture that the sequence is
                  finite: if p, p+2 is the largest twin prime pair generated, then test all n's
                  up to n = p. Obviously p! shares a factor with all twin primes in the set,
                  and so does n! for all n > p. Thus, no new twin prime pairs can be generated
                  from then on. Of course, exhausting the range up to n! when n is in the
                  thousands of digits isn't exactly practical.


                  [Non-text portions of this message have been removed]
                • Milton Brown
                  Even a computer search does not help you! (3629027, 3629029) = (10!+227, 10!+229) Unless n=10 means something else to you. (Maybe its not the computer after
                  Message 8 of 13 , May 5, 2005
                  View Source
                  • 0 Attachment
                    Even a computer search does not help you!

                    (3629027, 3629029) = (10!+227, 10!+229)

                    Unless n=10 means something else to you.

                    (Maybe its not the computer after all)

                    Milton L. Brown

                    > [Original Message]
                    > From: D�cio Luiz Gazzoni Filho <decio@...>
                    > To: <primenumbers@yahoogroups.com>
                    > Date: 5/5/2005 8:06:20 PM
                    > Subject: Re: [PrimeNumbers] Infinite Number of Twin Primes
                    >
                    > On Thursday 05 May 2005 20:13, you wrote:
                    > > Finally, 7! doesn't produce any values. And thus Milton's conjecture is
                    > > demolished, as about anything that he brainfarts on this list.
                    >
                    > Moreover, a PARI/GP search plus a clever shell script using PFGW (guess I
                    > should learn PFGW's scripting language instead, but...) confirms my
                    previous
                    > findings, and adds failures at n! for n = 10, 12-14, 16, 17, 22-64,
                    66-70,
                    > 72-135, 137-361, 363-683,685-900; that's as far as I'm willing to take
                    this
                    > search. I have reason to believe that only finitely many twin prime pairs
                    are
                    > generated by this process.
                    >
                    > Additionally, in the course of running this code, I discovered a bug in
                    > PARI/GP 2.2.9:
                    >
                    > ? isprime(42542905343533366778773944705953203289361426945380795183689)
                    > *** isprime: impossible inverse modulo: Mod(0,
                    > 42542905343533366778773944705953203289361426945380795183689).
                    >
                    > Ironically, Milton's work was not 100% in vain.
                    >
                    > D�cio
                    >
                    > PS: in principle one can actually check my conjecture that the sequence
                    is
                    > finite: if p, p+2 is the largest twin prime pair generated, then test all
                    n's
                    > up to n = p. Obviously p! shares a factor with all twin primes in the
                    set,
                    > and so does n! for all n > p. Thus, no new twin prime pairs can be
                    generated
                    > from then on. Of course, exhausting the range up to n! when n is in the
                    > thousands of digits isn't exactly practical.
                    >
                    >
                    > [Non-text portions of this message have been removed]
                    >
                    >
                    >
                    >
                    > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
                    > The Prime Pages : http://www.primepages.org/
                    >
                    >
                    > Yahoo! Groups Links
                    >
                    >
                    >
                    >
                    >
                  • Décio Luiz Gazzoni Filho
                    ... Besides being a moron, are you also unable to read? Here, let me reproduce my ... What part of this don t you understand? Just replace (59,61) by
                    Message 9 of 13 , May 5, 2005
                    View Source
                    • 0 Attachment
                      On Friday 06 May 2005 01:04, Milton Brown wrote:
                      > Even a computer search does not help you!
                      >
                      > (3629027, 3629029) = (10!+227, 10!+229)
                      >
                      > Unless n=10 means something else to you.
                      >
                      > (Maybe its not the computer after all)

                      Besides being a moron, are you also unable to read? Here, let me reproduce my
                      reply to your prior `objection':

                      > From your message that started the thread:
                      >
                      > > 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.
                      >
                      > Emphasis on `prior pairs'. Now tell me which values of n,k generate (n! + k,
                      > n! + k + 2) = (59,61).

                      What part of this don't you understand? Just replace (59,61) by (227,229). I
                      think I should write a shell script to automatically answer your emails.

                      I once wrote an opinion piece for a computer magazine about how computers
                      should require a license to operate, akin to a driver's license. My argument
                      at the time was that computer-illiterate users were polluting the internet
                      with viruses, trojan/worm traffic (not to mention stupid rehashed jokes,
                      chain letters, hoaxes and the like). Guess I should have extended the
                      proposal to morons with IQ < 50 and serious learning/reading/comprehension
                      and psychological issues, which waste other people's time through posts like
                      this on newsgroups and forums. Milton Brown and JSH from sci.math would be
                      the perfect posterchilds for such a proposal.

                      Why don't you do a favor to humanity and throw yourself off a tall building?

                      Décio


                      [Non-text portions of this message have been removed]
                    • Milton Brown
                      More addition for your computer: (479001791, 479001793) = (12!+191, 12!+193) And, again you said 12 didn t work! Milton L. Brown ... reproduce my ... of ...
                      Message 10 of 13 , May 6, 2005
                      View Source
                      • 0 Attachment
                        More addition for your computer:

                        (479001791, 479001793) = (12!+191, 12!+193)

                        And, again you said 12 didn't work!

                        Milton L. Brown


                        > [Original Message]
                        > From: D�cio Luiz Gazzoni Filho <decio@...>
                        > To: <primenumbers@yahoogroups.com>
                        > Date: 5/5/2005 9:35:46 PM
                        > Subject: Re: [PrimeNumbers] Infinite Number of Twin Primes
                        >
                        > On Friday 06 May 2005 01:04, Milton Brown wrote:
                        > > Even a computer search does not help you!
                        > >
                        > > (3629027, 3629029) = (10!+227, 10!+229)
                        > >
                        > > Unless n=10 means something else to you.
                        > >
                        > > (Maybe its not the computer after all)
                        >
                        > Besides being a moron, are you also unable to read? Here, let me
                        reproduce my
                        > reply to your prior `objection':
                        >
                        > > From your message that started the thread:
                        > >
                        > > > 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.
                        > >
                        > > Emphasis on `prior pairs'. Now tell me which values of n,k generate (n!
                        + k,
                        > > n! + k + 2) = (59,61).
                        >
                        > What part of this don't you understand? Just replace (59,61) by
                        (227,229). I
                        > think I should write a shell script to automatically answer your emails.
                        >
                        > I once wrote an opinion piece for a computer magazine about how computers
                        > should require a license to operate, akin to a driver's license. My
                        argument
                        > at the time was that computer-illiterate users were polluting the
                        internet
                        > with viruses, trojan/worm traffic (not to mention stupid rehashed jokes,
                        > chain letters, hoaxes and the like). Guess I should have extended the
                        > proposal to morons with IQ < 50 and serious
                        learning/reading/comprehension
                        > and psychological issues, which waste other people's time through posts
                        like
                        > this on newsgroups and forums. Milton Brown and JSH from sci.math would
                        be
                        > the perfect posterchilds for such a proposal.
                        >
                        > Why don't you do a favor to humanity and throw yourself off a tall
                        building?
                        >
                        > D�cio
                        >
                        >
                        > [Non-text portions of this message have been removed]
                        >
                        >
                        >
                        >
                        > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
                        > The Prime Pages : http://www.primepages.org/
                        >
                        >
                        > Yahoo! Groups Links
                        >
                        >
                        >
                        >
                        >
                      • Jacques Tramu
                        How many fact-twins are there ? A pair of fact-twins are twins primes (ft, ft+2) if exists n such as ft = n! + p, where p and p+2 are twin primes; The
                        Message 11 of 13 , May 6, 2005
                        View Source
                        • 0 Attachment
                          How many fact-twins are there ?

                          A pair of fact-twins are twins primes (ft, ft+2) if exists n such as
                          ft = n! + p, where p and p+2 are twin primes;

                          The following table count the number of fact-twins in intervals of length
                          10^9 starting at
                          0, up to 10^18.

                          pi = number of primes in the interval [from, from+10^9]
                          twins = number of twins
                          fact-twins : number of fact-twins
                          % = fact-twins * 100 / twins

                          from pi twins fact-twins
                          %
                          1 50847534 3424506 999836 29,20%
                          10^9 47374753 2963535 815419 27,51%
                          10^10 43336106 2477174 674611 27,23%
                          10^11 39475591 2055627 547106 26,62%
                          10^12 36190991 1730012 386360 22,33%
                          10^13 33405006 1473196 318563 21,62%
                          10^14 31019409 1270499 265221 20,88%
                          10^15 28946421 1105560 225366 20,38%
                          10^16 27153205 972510 194566 20,01%
                          10^17 25549226 861742 153809 17,85%
                          10^18 24127085 769103 135977 17,68%
                        • Jud McCranie
                          ... the point is that you gave this procedure for producing more and more twin primes from factorials and the previously found twin primes. Other than a
                          Message 12 of 13 , May 6, 2005
                          View Source
                          • 0 Attachment
                            At 03:09 AM 5/6/2005, Milton Brown wrote:
                            >More addition for your computer:
                            >
                            > (479001791, 479001793) = (12!+191, 12!+193)
                            >
                            >And, again you said 12 didn't work!

                            the point is that you gave this procedure for producing more and more twin
                            primes from factorials and the previously found twin primes. Other than a
                            finite number of twin primes as seeds and the ones previously generated by
                            your procedure, your procedure doesn't know any other twin primes. The
                            procedure has to bootstrap itself and keep going forever. IIRC, he showed
                            that 191 and 193 can't be generated that way, so you can't use them for 12!.
                          • Mark Underwood
                            Jud s point is well taken. Milton s original offering was that twin primes would be generated from only previously generated twin primes. I took the liberty to
                            Message 13 of 13 , May 6, 2005
                            View Source
                            • 0 Attachment
                              Jud's point is well taken. Milton's original offering was that twin
                              primes would be generated from only previously generated twin primes.

                              I took the liberty to modify Milton's original offering to this:

                              Starting with n=3 and (5,7) as the first twin prime pair, if we add
                              *or subtract* a (previously generated) twin prime pair from n! to
                              generate more twin prime pairs, how far can n go before there is a
                              failure to produce twins? Answer : n=22.

                              But if instead of n! we use p#, then we can get as high as p = 61
                              before there is a failure to produce. (61 is the only prime to fail
                              before 103. After that failures become more common, as one might
                              expect.)


                              Mark


                              PS to certain others: Where is the love? How can there ever be peace
                              in the world if we respect our own knowledge or sensibilities or
                              tradition over our fellow human beings?
                            Your message has been successfully submitted and would be delivered to recipients shortly.