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2^p+3

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  • Paul Jobling
    Hi, I was just wondering what the state of play was with looking for a (pseudo-)prime of the form 2^p+3 - what search limits have been reached, and have any
    Message 1 of 22 , May 1, 2002
      Hi,

      I was just wondering what the state of play was with looking for a
      (pseudo-)prime of the form 2^p+3 - what search limits have been reached, and
      have any PRP's been found? I recall that there was some searching being done a
      couple of years ago (I think), but I do not know what (if any) results were
      found?

      Regards,

      Paul.


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    • rlberry2002
      Paul, Just a couple of observations regarding primes of the form 2^p+3. First, there should be infinitely many primes of this form. When p is even, the result
      Message 2 of 22 , May 1, 2002
        Paul,

        Just a couple of observations regarding primes of the form 2^p+3.
        First, there should be infinitely many primes of this form. When p
        is even, the result is a number congruent 1 Mod 6; when p is odd, the
        result is a number congruent 5 Mod 6. Since all primes other than 2
        or 3 are congruent 1 Mod 6 or 5 Mod 6 and since there are infinitely
        many primes contained in either of the two congruences, it follows
        that there should be infinitely many primes of the form 2^p+3. I
        haven't a clue as to the largest prime to date of this form, but
        certainly it should be (or could be, with a little focused attention)
        very large.

        The prime number form you propose are very similar to Mersenne primes
        and I would expect prime number results for 2^p+3 to rival those of
        Mersenne primes in size and distribution too.

        Robert

        --- In primenumbers@y..., "Paul Jobling" <Paul.Jobling@W...> wrote:
        > Hi,
        >
        > I was just wondering what the state of play was with looking for a
        > (pseudo-)prime of the form 2^p+3 - what search limits have been
        reached, and
        > have any PRP's been found? I recall that there was some searching
        being done a
        > couple of years ago (I think), but I do not know what (if any)
        results were
        > found?
        >
        > Regards,
        >
        > Paul.
        >
        >
        > __________________________________________________
        > Virus checked by MessageLabs Virus Control Centre.
      • Paul Jobling
        Hi Robert, ... I believe that we are only interested in p prime here. ... Careful... that sort of reasoning is wrong - you are saying that given an infinite
        Message 3 of 22 , May 1, 2002
          Hi Robert,

          > Just a couple of observations regarding primes of the form 2^p+3.
          > First, there should be infinitely many primes of this form. When p
          > is even, the result is a number congruent 1 Mod 6; when p is odd, the
          > result is a number congruent 5 Mod 6.

          I believe that we are only interested in p prime here.

          > Since all primes other than 2
          > or 3 are congruent 1 Mod 6 or 5 Mod 6 and since there are infinitely
          > many primes contained in either of the two congruences, it follows
          > that there should be infinitely many primes of the form 2^p+3.

          Careful... that sort of reasoning is wrong - you are saying that given an
          infinite set N, where an infinite number of its member have some property A,
          then any infinite subset M of N must contain an infinite number of members
          with the proper A as well (consider N=the integers; A=prime; M=the composite
          numbers).

          > I
          > haven't a clue as to the largest prime to date of this form, but
          > certainly it should be (or could be, with a little focused attention)
          > very large.
          >
          > The prime number form you propose are very similar to Mersenne primes
          > and I would expect prime number results for 2^p+3 to rival those of
          > Mersenne primes in size and distribution too.

          But they do not seem to. The earliest Mersenne primes are quite small, whereas
          I am not sure that even one example of a pseudoprime of this form has been
          found.

          Regards,

          Paul.


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        • Paul Jobling
          ... Let me correct that to say for reasonably large p . __________________________________________________ Virus checked by MessageLabs Virus Control Centre.
          Message 4 of 22 , May 1, 2002
            > But they do not seem to. The earliest Mersenne primes are
            > quite small, whereas
            > I am not sure that even one example of a pseudoprime of this
            > form has been found.

            Let me correct that to say "for reasonably large p".

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          • paulunderwooduk
            ... reached, and ... being done a ... results were ... http://groups.yahoo.com/group/primenumbers/message/1023 might contain some results for 2^p+3 Paul U.
            Message 5 of 22 , May 1, 2002
              --- In primenumbers@y..., "Paul Jobling" <Paul.Jobling@W...> wrote:
              > Hi,
              >
              > I was just wondering what the state of play was with looking for a
              > (pseudo-)prime of the form 2^p+3 - what search limits have been
              reached, and
              > have any PRP's been found? I recall that there was some searching
              being done a
              > couple of years ago (I think), but I do not know what (if any)
              results were
              > found?

              http://groups.yahoo.com/group/primenumbers/message/1023

              might contain some results for 2^p+3

              Paul U.
            • Paul Jobling
              Well, the OLEIS (A057736) only has 2, 3, 7, 67. But looking at http://groups.yahoo.com/group/primeform/message/1218 it appears that some searching was going
              Message 6 of 22 , May 1, 2002
                Well, the OLEIS (A057736) only has 2, 3, 7, 67. But looking at
                http://groups.yahoo.com/group/primeform/message/1218
                it appears that some searching was going on. So now the question is what
                search limits were reached - Christ; Chris?

                Paul.


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              • rlberry2002
                Paul, I obviously misunderstood your equation; I read it as (2^p) + 3 where p=2,result 7; p=3, result 11, and so on. I stand by my earlier statements (let s
                Message 7 of 22 , May 1, 2002
                  Paul,

                  I obviously misunderstood your equation; I read it as (2^p) + 3 where
                  p=2,result 7; p=3, result 11, and so on.

                  I stand by my earlier statements (let's term it as Robert's
                  Conjecture) though: A one variable equation which, (a) does not
                  reduce and (b) contains infinitely many 1 Mod 6 and/or 5 Mod 6
                  numbers, will contain infinitely many prime numbers.

                  I enjoy your comments and perspective...

                  Robert

                  --- In primenumbers@y..., "Paul Jobling" <Paul.Jobling@W...> wrote:
                  > Hi Robert,
                  >
                  > > Just a couple of observations regarding primes of the form 2^p+3.
                  > > First, there should be infinitely many primes of this form. When
                  p
                  > > is even, the result is a number congruent 1 Mod 6; when p is odd,
                  the
                  > > result is a number congruent 5 Mod 6.
                  >
                  > I believe that we are only interested in p prime here.
                  >
                  > > Since all primes other than 2
                  > > or 3 are congruent 1 Mod 6 or 5 Mod 6 and since there are
                  infinitely
                  > > many primes contained in either of the two congruences, it follows
                  > > that there should be infinitely many primes of the form 2^p+3.
                  >
                  > Careful... that sort of reasoning is wrong - you are saying that
                  given an
                  > infinite set N, where an infinite number of its member have some
                  property A,
                  > then any infinite subset M of N must contain an infinite number of
                  members
                  > with the proper A as well (consider N=the integers; A=prime; M=the
                  composite
                  > numbers).
                  >
                  > > I
                  > > haven't a clue as to the largest prime to date of this form, but
                  > > certainly it should be (or could be, with a little focused
                  attention)
                  > > very large.
                  > >
                  > > The prime number form you propose are very similar to Mersenne
                  primes
                  > > and I would expect prime number results for 2^p+3 to rival those
                  of
                  > > Mersenne primes in size and distribution too.
                  >
                  > But they do not seem to. The earliest Mersenne primes are quite
                  small, whereas
                  > I am not sure that even one example of a pseudoprime of this form
                  has been
                  > found.
                  >
                  > Regards,
                  >
                  > Paul.
                  >
                  >
                  > __________________________________________________
                  > Virus checked by MessageLabs Virus Control Centre.
                • paulunderwooduk
                  Paul, btw this ABC2 2^($a)-2^($a/2+1)+1 a: from 2 to 3000 produced: 2^3-2^(3/2+1)+1 2^7-2^(7/2+1)+1 2^47-2^(47/2+1)+1 2^73-2^(73/2+1)+1 2^79-2^(79/2+1)+1
                  Message 8 of 22 , May 1, 2002
                    Paul,

                    btw this

                    ABC2 2^($a)-2^($a/2+1)+1
                    a: from 2 to 3000

                    produced:

                    2^3-2^(3/2+1)+1
                    2^7-2^(7/2+1)+1
                    2^47-2^(47/2+1)+1
                    2^73-2^(73/2+1)+1
                    2^79-2^(79/2+1)+1
                    2^113-2^(113/2+1)+1
                    2^151-2^(151/2+1)+1
                    2^167-2^(167/2+1)+1
                    2^239-2^(239/2+1)+1
                    2^241-2^(241/2+1)+1
                    2^353-2^(353/2+1)+1
                    2^367-2^(367/2+1)+1
                    2^457-2^(457/2+1)+1
                    2^1367-2^(1367/2+1)+1

                    Note the first exponents are all prime. Are there anymore of these?

                    Paul U.
                  • Phil Carmody
                    ... [SNIP - see Paul J s reply] ... I wouldn t expect them to have the same distribution. 2^p+1, being a cyclotomic form, has restrictions on what its factors
                    Message 9 of 22 , May 1, 2002
                      --- rlberry2002 <rlberry2002@...> wrote:
                      > Just a couple of observations regarding primes of the form 2^p+3.

                      [SNIP - see Paul J's reply]

                      > The prime number form you propose are very similar to Mersenne
                      > primes
                      > and I would expect prime number results for 2^p+3 to rival those of
                      >
                      > Mersenne primes in size and distribution too.

                      I wouldn't expect them to have the same distribution.
                      2^p+1, being a cyclotomic form, has restrictions on what its factors
                      can be. 2^p+3 has no such divisibility criterea.

                      Phil

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                    • Chris Caldwell
                      ... Such as: 2^364289-2^182145+1 Sure. These are norms of the Gaussian Mersenne primes. Look on the prime list.
                      Message 10 of 22 , May 1, 2002
                        At 04:02 PM 5/1/02 +0000, paulunderwooduk wrote:
                        >2^167-2^(167/2+1)+1
                        >2^239-2^(239/2+1)+1
                        >2^241-2^(241/2+1)+1
                        >2^353-2^(353/2+1)+1
                        >2^367-2^(367/2+1)+1
                        >2^457-2^(457/2+1)+1
                        >2^1367-2^(1367/2+1)+1
                        >
                        >Note the first exponents are all prime. Are there anymore of these?

                        Such as:

                        2^364289-2^182145+1

                        Sure. These are norms of the Gaussian Mersenne primes. Look on the
                        prime list.
                      • jbrennen
                        ... No, you got it right :-) ... See Sierpinski numbers for a well-known counterexample: 78557*2^n+1 contains an infinite number of elements which are (5 mod
                        Message 11 of 22 , May 1, 2002
                          --- In primenumbers@y..., "rlberry2002" <rlberry2002@y...> wrote:
                          > Paul,
                          >
                          > I obviously misunderstood your equation; I read it as (2^p) + 3
                          > where p=2,result 7; p=3, result 11, and so on.

                          No, you got it right :-)

                          > I stand by my earlier statements (let's term it as Robert's
                          > Conjecture) though: A one variable equation which, (a) does not
                          > reduce and (b) contains infinitely many 1 Mod 6 and/or 5 Mod 6
                          > numbers, will contain infinitely many prime numbers.

                          See Sierpinski numbers for a well-known counterexample:
                          78557*2^n+1 contains an infinite number of elements which are
                          (5 mod 6), but has no prime numbers for any integer n.

                          Back to the original question... There are easily found concrete
                          examples of the form 2^p+n which do not have an infinite number of
                          prime values (with p prime) despite having an infinite number of
                          (1 mod 6) and (5 mod 6) numbers:

                          N=2^p+12213 (with p prime) has no prime values whatsoever.

                          This is because:

                          If p == 2, N is divisible by 19
                          If p == 3, N is divisible by 11
                          If p == 1 (mod 12), N is divisible by 5
                          If p == 5 (mod 12), N is divisible by 5
                          If p == 7 (mod 12), N is divisible by 7
                          If p == 11 (mod 12), N is divisible by 13

                          Every prime p meets one of the six cases above, so N is never
                          prime when p is prime.
                        • Phil Carmody
                          ... They of course have their own divisibility criterea. But one unrelated to (a^n+b^n) forms. Their criteria are more similar to those behind the fixed k
                          Message 12 of 22 , May 1, 2002
                            --- Phil Carmody <thefatphil@...> wrote:
                            > > The prime number form you propose are very similar to Mersenne
                            > > primes
                            > > and I would expect prime number results for 2^p+3 to rival those
                            > of
                            > >
                            > > Mersenne primes in size and distribution too.
                            >
                            > I wouldn't expect them to have the same distribution.
                            > 2^p+1, being a cyclotomic form, has restrictions on what its
                            > factors
                            > can be. 2^p+3 has no such divisibility criterea.

                            They of course have their own divisibility criterea. But one
                            unrelated to (a^n+b^n) forms. Their criteria are more similar to
                            those behind the 'fixed k Proth' problems.

                            Phil

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                          • jim_fougeron
                            ... For an infinity of easy to show counter examples, look at this form: k#+p If p is a fixed prime 3, then the expression will be +-1 mod(6), however, the
                            Message 13 of 22 , May 1, 2002
                              --- In primenumbers@y..., "jbrennen" <jack@b...> wrote:
                              >--- In primenumbers@y..., "rlberry2002" <rlberry2002@y...> wrote:
                              >> Paul,
                              >>
                              >> I obviously misunderstood your equation; I read it as (2^p) + 3
                              >> where p=2,result 7; p=3, result 11, and so on.
                              >
                              >No, you got it right :-)
                              >
                              >> I stand by my earlier statements (let's term it as Robert's
                              >> Conjecture) though: A one variable equation which, (a) does not
                              >> reduce and (b) contains infinitely many 1 Mod 6 and/or 5 Mod 6
                              >> numbers, will contain infinitely many prime numbers.
                              >
                              > See Sierpinski numbers for a well-known counterexample:
                              > 78557*2^n+1 contains an infinite number of elements which are
                              > (5 mod 6), but has no prime numbers for any integer n.

                              For an infinity of easy to show counter examples, look at this form:

                              k#+p

                              If p is a "fixed" prime > 3, then the expression will be +-1 mod(6),
                              however, the expression (for a variable k and fixed p) will produce
                              ONLY a finite (if any) amount of primes. Primes can only be generated
                              by the above form while k < p. Once k reaches the size of p, then
                              p will always be a factor of the expression.

                              Take for example k#+13.
                              This is prime for k=3, 5, 7 and NO others, since 2#+13=3*5,
                              11#+13= 23*101 and when k>=13 then k#+13 is always has a factor of 13.
                              However k#+13 == 1mod(6) (except for 2#+13==3mod(6)).

                              Jim.
                            • mikeoakes2@aol.com
                              AFAIK the largest PRP of form 2^n+3 is still the one found by me in July 2001:- 2^122550+3 It is the 19th largest known PRP, according to Henri Lifchitz s
                              Message 14 of 22 , May 1, 2002
                                AFAIK the largest PRP of form 2^n+3 is still the one found by me in July
                                2001:-
                                2^122550+3

                                It is the 19th largest known PRP, according to Henri Lifchitz's database at
                                http://ourworld.compuserve.com/homepages/hlifchitz/

                                The sequence for lower values is Sloane's A057732. I had searched up to
                                n=127677, and proved primality for n <=2370 using Titanix, finishing this
                                project on 15 Aug 2001.

                                (See also my post to the primenumbers group dated 8 Jul 2002.)

                                Mike Oakes


                                In a message dated 01/05/02 14:41:02 GMT Daylight Time,
                                Paul.Jobling@... writes:

                                > Hi,
                                >
                                > I was just wondering what the state of play was with looking for a
                                > (pseudo-)prime of the form 2^p+3 - what search limits have been reached,
                                > and
                                > have any PRP's been found? I recall that there was some searching being
                                > done a
                                > couple of years ago (I think), but I do not know what (if any) results were
                                > found?
                                >
                                > Regards,
                                >
                                > Paul.
                                >




                                [Non-text portions of this message have been removed]
                              • rlberry2002
                                Please pardon my ignorance; I am here to learn to grow in my knowledge of number theory. However, I will try to be more careful in responding to posts before
                                Message 15 of 22 , May 1, 2002
                                  Please pardon my ignorance; I am here to learn to grow in my
                                  knowledge of number theory. However, I will try to be more careful
                                  in responding to posts before I have fully thought out my response -
                                  you could extend me the same courtesy.

                                  Your example of N=2^p + 12213 violates one of the two qualifications
                                  that I laid down - "the equation cannot be reducible". I typically
                                  would use this tenet for an equation like 4n + 1 (which does have
                                  infinitely many primes in it)where n is odd. The tenet holds for
                                  numbers of the form 2^p + n also, it just is usually harder to find
                                  the pattern that this type of equation reduces to. In your example,
                                  you provide the pattern which violates my first condition: that is
                                  for 2, 3, 1 Mod 12, 5 Mod 12, 7 Mod 12, & 11 Mod 12; there a fixed
                                  set of possible outcomes each of which will have fixed factors
                                  depending upon which of the outcomes it fall under.

                                  For instance, the equation 30Y + 35 = N generates infinitely many 5
                                  Mod 6 numbers none of which are prime. This equation is easy since it
                                  reduces to 5*(6Y + 7) = N; numbers of the form 2^p + N require a
                                  deeper analysis as long as N itself does not have a factor of 2^p.

                                  Just a few thoughts

                                  Robert

                                  --- In primenumbers@y..., "jbrennen" <jack@b...> wrote:
                                  > --- In primenumbers@y..., "rlberry2002" <rlberry2002@y...> wrote:
                                  > > Paul,
                                  > >
                                  > > I obviously misunderstood your equation; I read it as (2^p) + 3
                                  > > where p=2,result 7; p=3, result 11, and so on.
                                  >
                                  > No, you got it right :-)
                                  >
                                  > > I stand by my earlier statements (let's term it as Robert's
                                  > > Conjecture) though: A one variable equation which, (a) does not
                                  > > reduce and (b) contains infinitely many 1 Mod 6 and/or 5 Mod 6
                                  > > numbers, will contain infinitely many prime numbers.
                                  >
                                  > See Sierpinski numbers for a well-known counterexample:
                                  > 78557*2^n+1 contains an infinite number of elements which are
                                  > (5 mod 6), but has no prime numbers for any integer n.
                                  >
                                  > Back to the original question... There are easily found concrete
                                  > examples of the form 2^p+n which do not have an infinite number of
                                  > prime values (with p prime) despite having an infinite number of
                                  > (1 mod 6) and (5 mod 6) numbers:
                                  >
                                  > N=2^p+12213 (with p prime) has no prime values whatsoever.
                                  >
                                  > This is because:
                                  >
                                  > If p == 2, N is divisible by 19
                                  > If p == 3, N is divisible by 11
                                  > If p == 1 (mod 12), N is divisible by 5
                                  > If p == 5 (mod 12), N is divisible by 5
                                  > If p == 7 (mod 12), N is divisible by 7
                                  > If p == 11 (mod 12), N is divisible by 13
                                  >
                                  > Every prime p meets one of the six cases above, so N is never
                                  > prime when p is prime.
                                • Phil Carmody
                                  ... I would trust that such courtesy is demonstrated on the list. (Yeah, flame me off list, you won t be the first... (or the second) ... Strictly, it is
                                  Message 16 of 22 , May 1, 2002
                                    --- rlberry2002 <rlberry2002@...> wrote:
                                    > Please pardon my ignorance; I am here to learn to grow in my
                                    > knowledge of number theory. However, I will try to be more careful
                                    > in responding to posts before I have fully thought out my response
                                    > - you could extend me the same courtesy.

                                    I would trust that such courtesy is demonstrated on the list.
                                    (Yeah, flame me off list, you won't be the first... (or the second)
                                    :-) )

                                    > Your example of N=2^p + 12213 violates one of the two
                                    > qualifications
                                    > that I laid down - "the equation cannot be reducible".

                                    Strictly, it is irreducible. The letter of the law is obeyed.

                                    > I typically
                                    > would use this tenet for an equation like 4n + 1 (which does have
                                    > infinitely many primes in it)where n is odd. The tenet holds for
                                    > numbers of the form 2^p + n also, it just is usually harder to find
                                    >
                                    > the pattern that this type of equation reduces to. In your
                                    > example,
                                    > you provide the pattern which violates my first condition: that is
                                    >
                                    > for 2, 3, 1 Mod 12, 5 Mod 12, 7 Mod 12, & 11 Mod 12; there a fixed
                                    > set of possible outcomes each of which will have fixed factors
                                    > depending upon which of the outcomes it fall under.

                                    Sure, but that's not redicibility. What Jack has highlighted is an
                                    intrinsically interesting property about that sequence of numbers.
                                    This property will ba shared by an infinite number of other
                                    sequences, not just the ...+12213. The property isn't reducibility,
                                    and that term was used, it's a very precisely defined term, so Jack
                                    and others can't be faulted for taking the term at face falue.
                                    Perhaps the term "with no intrinsic predicatble factorisations" or
                                    similar could be used to cover such concepts.

                                    Such forms are certainly vastly interesting, the Sierpinski/Riesel
                                    problems (that Jack mentioned, IIRC) are all about whether there are
                                    primes with forms that have no intricsic predictable factorisations.

                                    (hmmm, reminder to self or others - is there a link waiting to be
                                    added to the yahoogroup regarding the Sierpinski/Riesel problems?)

                                    Phil

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                                  • rlberry2002
                                    Phil, As always, your comments and insights are welcome. I had to do a little refresher myself on Sierpinski numbers: a positive, odd integer k for which
                                    Message 17 of 22 , May 1, 2002
                                      Phil,

                                      As always, your comments and insights are welcome. I had to do a
                                      little refresher myself on Sierpinski numbers: a positive, odd
                                      integer k for which integers of the form k*2^p + 1 are all composite.
                                      I would suggest that there is little here which serves as an adequate
                                      counter example to the conjecture that I previously made.

                                      First, for all k less than 78557, at least 1 prime has been found to
                                      be generated by k*2^p + 1 with only 19 exceptions (and I feel it is
                                      just that the first prime solution for these 19 k's has not yet been
                                      found). It is only conjectured that k=78557 is a Sierpinski number.
                                      It is likely that the smallest Sierpinski number (if indeed one does
                                      exist) is so large that direct factorization, indirect factorization,
                                      etc. will not be feasible. One final point concerning k=78557 will
                                      show the difficulty in analyzing these numbers: For k=78557, p=300
                                      you get a 95-digit result. In other words, the 300th example for
                                      k=78557 is already a number of such magnitude that only 1 number in
                                      300 will be prime. Guess what the odds look like for the next 300
                                      values of p.

                                      Indeed, any series of numbers of the form k*N^p +/- c are very
                                      difficult to analysis except with indirect methods.

                                      Regards,

                                      Robert

                                      --- In primenumbers@y..., Phil Carmody <thefatphil@y...> wrote:
                                      > --- rlberry2002 <rlberry2002@y...> wrote:
                                      > > Please pardon my ignorance; I am here to learn to grow in my
                                      > > knowledge of number theory. However, I will try to be more
                                      careful
                                      > > in responding to posts before I have fully thought out my response
                                      > > - you could extend me the same courtesy.
                                      >
                                      > I would trust that such courtesy is demonstrated on the list.
                                      > (Yeah, flame me off list, you won't be the first... (or the second)
                                      > :-) )
                                      >
                                      > > Your example of N=2^p + 12213 violates one of the two
                                      > > qualifications
                                      > > that I laid down - "the equation cannot be reducible".
                                      >
                                      > Strictly, it is irreducible. The letter of the law is obeyed.
                                      >
                                      > > I typically
                                      > > would use this tenet for an equation like 4n + 1 (which does have
                                      > > infinitely many primes in it)where n is odd. The tenet holds for
                                      > > numbers of the form 2^p + n also, it just is usually harder to
                                      find
                                      > >
                                      > > the pattern that this type of equation reduces to. In your
                                      > > example,
                                      > > you provide the pattern which violates my first condition: that
                                      is
                                      > >
                                      > > for 2, 3, 1 Mod 12, 5 Mod 12, 7 Mod 12, & 11 Mod 12; there a
                                      fixed
                                      > > set of possible outcomes each of which will have fixed factors
                                      > > depending upon which of the outcomes it fall under.
                                      >
                                      > Sure, but that's not redicibility. What Jack has highlighted is an
                                      > intrinsically interesting property about that sequence of numbers.
                                      > This property will ba shared by an infinite number of other
                                      > sequences, not just the ...+12213. The property isn't reducibility,
                                      > and that term was used, it's a very precisely defined term, so Jack
                                      > and others can't be faulted for taking the term at face falue.
                                      > Perhaps the term "with no intrinsic predicatble factorisations" or
                                      > similar could be used to cover such concepts.
                                      >
                                      > Such forms are certainly vastly interesting, the Sierpinski/Riesel
                                      > problems (that Jack mentioned, IIRC) are all about whether there are
                                      > primes with forms that have no intricsic predictable factorisations.
                                      >
                                      > (hmmm, reminder to self or others - is there a link waiting to be
                                      > added to the yahoogroup regarding the Sierpinski/Riesel problems?)
                                      >
                                      > Phil
                                      >
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                                    • Jack Brennen
                                      ... We re glad you did a little research on Sierpinski numbers. However, you must have missed something. It is PROVEN, and can be shown using nothing more
                                      Message 18 of 22 , May 1, 2002
                                        rlberry2002 wrote:
                                        > As always, your comments and insights are welcome. I had to do a
                                        > little refresher myself on Sierpinski numbers: a positive, odd
                                        > integer k for which integers of the form k*2^p + 1 are all composite.
                                        > I would suggest that there is little here which serves as an adequate
                                        > counter example to the conjecture that I previously made.
                                        >
                                        > First, for all k less than 78557, at least 1 prime has been found to
                                        > be generated by k*2^p + 1 with only 19 exceptions (and I feel it is
                                        > just that the first prime solution for these 19 k's has not yet been
                                        > found). It is only conjectured that k=78557 is a Sierpinski number.

                                        We're glad you did a little research on Sierpinski numbers.

                                        However, you must have missed something. It is PROVEN, and can be shown
                                        using nothing more than very simple arithmetic, that k=78557 is
                                        a Sierpinski number. The proof, in condensed form:

                                        If n == 0 (mod 2), 78557*2^n+1 is divisible by 3
                                        If n == 1 (mod 4), 78557*2^n+1 is divisible by 5
                                        If n == 3 (mod 36), 78557*2^n+1 is divisible by 73
                                        If n == 15 (mod 36), 78557*2^n+1 is divisible by 19
                                        If n == 27 (mod 36), 78557*2^n+1 is divisible by 37
                                        If n == 7 (mod 12), 78557*2^n+1 is divisible by 7
                                        If n == 11 (mod 12), 78557*2^n+1 is divisible by 13

                                        Every integer n satisfies one of these seven congruences.

                                        The unproven conjecture is that k=78557 is the
                                        *smallest* Sierpinski number.
                                      • Gary Chaffey
                                        Does the idea of 2^p+3 extend to 2^p-3? I have found that 2^233-3 is PRP now p is of the form 60k-7. Is this just a coincedence or is there some sort of
                                        Message 19 of 22 , May 3, 2002
                                          Does the idea of 2^p+3 extend to 2^p-3? I have found
                                          that 2^233-3 is PRP now p is of the form 60k-7. Is
                                          this just a coincedence or is there some sort of
                                          pattern.
                                          I haven't checked
                                          2^233= a mod 233
                                          2^(2^233)= 2^n mod a
                                          Like Norman did for 2^p+3 with p=7 and 67
                                          Gary

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                                        • Phil Carmody
                                          ... This can be checked as follows. If 5 | 2^x-3 then 2^x==3 (5) then x==3 (4) If 7 | 2^x-3 then 2^x==3 (7) no solution If 11 | 2^x-3 then 2^x==3 (11) then
                                          Message 20 of 22 , May 3, 2002
                                            --- Gary Chaffey <garychaffey@...> wrote:
                                            > Does the idea of 2^p+3 extend to 2^p-3? I have found
                                            > that 2^233-3 is PRP now p is of the form 60k-7. Is
                                            > this just a coincedence or is there some sort of
                                            > pattern.

                                            This can be checked as follows.

                                            If 5 | 2^x-3 then 2^x==3 (5) then x==3 (4)
                                            If 7 | 2^x-3 then 2^x==3 (7) no solution
                                            If 11 | 2^x-3 then 2^x==3 (11) then x==8 (10)
                                            If 13 | 2^x-3 then 2^x==3 (13) then x==4 (12)
                                            ...

                                            These remove
                                            3,7,11,15,19,23,27,31,35,39,43,47,51,55,59 (mod 60)
                                            8 18 28 38 48 58 (mod 60)
                                            4 16 28 40 52 (mod 60)
                                            ...

                                            There are other primes that add to this mod 60 period, obviously.

                                            Not every residue is removed, which leads me to suspect that 53 isn't
                                            the only residue primes will be found along.

                                            Phil

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                                          • djbroadhurst
                                            A little reminder: if you find a PRP of the form 2^n-3 or 2^n+3 (for any n, not just a prime) you may have prospects of a BLS (which failing a KP) proof by
                                            Message 21 of 22 , May 3, 2002
                                              A little reminder: if you find a PRP of the form
                                              2^n-3 or 2^n+3 (for any n, not just a prime) you may
                                              have prospects of a BLS (which failing a KP) proof
                                              by looking at work on factorization of Phi(2,k),
                                              since, in *either* case, *both* N-1 and N+1 are
                                              algebraically factorizable into these intensively
                                              studied base-2 cyclotomic numbers:
                                              http://www.cerias.purdue.edu/homes/ssw/cun/index.html
                                              Apologies to those for whom this is blindingly obvious.
                                              David Broadhurst
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