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Re: 2^p+3

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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 1 of 22 , May 1, 2002
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      --- 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 2 of 22 , May 1, 2002
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        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 3 of 22 , May 1, 2002
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          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.
          >
          >
          > __________________________________________________
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        • 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 4 of 22 , May 1, 2002
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            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 5 of 22 , May 1, 2002
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              --- 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 6 of 22 , May 1, 2002
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                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 7 of 22 , May 1, 2002
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                  --- 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 8 of 22 , May 1, 2002
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                    --- 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 9 of 22 , May 1, 2002
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                      --- 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 10 of 22 , May 1, 2002
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                        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 11 of 22 , May 1, 2002
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                          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 12 of 22 , May 1, 2002
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                            --- 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 13 of 22 , May 1, 2002
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                              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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                              > Yahoo! Health - your guide to health and wellness
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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 14 of 22 , May 1, 2002
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                                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 15 of 22 , May 3, 2002
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                                  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 16 of 22 , May 3, 2002
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                                    --- 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 17 of 22 , May 3, 2002
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                                      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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