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(n+1)^p - n^p

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  • Mark Underwood
    It is well known that for the expression (n+1)^x - n^x to yield a prime number, x must itself be prime. (And if x is a prime p the same expression contains
    Message 1 of 10 , Mar 4, 2006
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      It is well known that for the expression

      (n+1)^x - n^x

      to yield a prime number, x must itself be prime.
      (And if x is a prime p the same expression contains only prime
      factors of the form p*2k + 1)

      Here's a question: For every prime p, is there always an n such that
      (n+1)^p - n^p is prime?

      I think so. Here are the lowest 15 values of n which yield a prime
      for the first 15 primes (p,n):

      (2,1) (3,1) (5,1) (7,1) (11,5) (13,1) (17,1) (19,1) (23,5) (29,2)
      (31,1) (37,39) (41,6) (43,4) (47,12)

      (For example (12+1)^47 - 12^47 is prime. )

      Here is a list (p,n) which shows only increasing n:

      (2,1) (11,5) (37,39) (73,46) (137,327) (239,367) (307,402) (419,781)
      (443,1774)

      (For example (1774+1)^443 - 1774^443 is prime. (Over 1400 digits, my
      biggest ever found , wheee! :) )

      Another question is this: Given a prime p is there an upper bound for
      n before which we can be sure to generate a prime?
      The value (1 + 1/e) crops up alot in my dabblings, so for fun I'll
      guess the following:

      (n+1)^p - n^p is prime for some n < p^(1+1/e)

      Oh the rigour of it all (not!)...

      Mark
    • Lelio sknet
      ... The following pfgw abc2 file ABC2 ($a+1)^$b-$a^$b // -f{$b} {number_primes,$b,1} a: from 1 to 10000 b: primes from 500 to 10000 with command pfgw –f
      Message 2 of 10 , Mar 4, 2006
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        > (For example (1774+1)^443 - 1774^443 is prime. (Over 1400 digits, my
        > biggest ever found , wheee! :) )

        > Mark

        The following pfgw abc2 file

        ABC2 ($a+1)^$b-$a^$b // -f{$b} {number_primes,$b,1}

        a: from 1 to 10000

        b: primes from 500 to 10000

        with command pfgw –f script.file can check your conjecture overnight up to
        prime 9973.
        In a few minutes I found (503,1105) (593,2175) (641,1469) (653,1167) and
        (683,1147) to be bigger than yours.

        Lélio







        [Non-text portions of this message have been removed]
      • Lelio sknet
        ... And according to Henri Lifchitz, the biggest ever found was (341233,3) which contains 205443 digits and was found by Jean Louis Charton last January. See
        Message 3 of 10 , Mar 4, 2006
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          > (For example (1774+1)^443 - 1774^443 is prime. (Over 1400 digits, my
          > biggest ever found , wheee! :) )

          > Mark

          > The following pfgw abc2 file

          > ABC2 ($a+1)^$b-$a^$b // -f{$b} {number_primes,$b,1}

          > a: from 1 to 10000

          > b: primes from 500 to 10000

          > with command pfgw –f script.file can check your conjecture overnight up to
          > prime 9973.
          > In a few minutes I found (503,1105) (593,2175) (641,1469) (653,1167) and
          > (683,1147) to be bigger than yours.

          > Lélio

          And according to Henri Lifchitz, the biggest ever found was (341233,3) which
          contains 205443 digits and was found by Jean Louis Charton last January.

          See Henri´s pages at



          http://www.primenumbers.net/prptop/prptop.php



          Lélio












          [Non-text portions of this message have been removed]
        • Wojciech.Florek@amu.edu.pl
          ... [snip] ... If I know, both, an ouput of pfgw (without -tc option) and Henri s pages, contain (strong, but) PRP s not primes. Mark has written about primes.
          Message 4 of 10 , Mar 4, 2006
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            On Sun, 5 Mar 2006, Lelio sknet wrote:

            > > Mark
            > > The following pfgw abc2 file
            >
            [snip]
            > > with command pfgw –f script.file can check your conjecture overnight up to
            > > prime 9973.
            > > In a few minutes I found (503,1105) (593,2175) (641,1469) (653,1167) and
            > > (683,1147) to be bigger than yours.
            >
            > > Lélio
            >
            > And according to Henri Lifchitz, the biggest ever found was (341233,3) which
            > contains 205443 digits and was found by Jean Louis Charton last January.
            >
            > See Henri´s pages at
            >
            > http://www.primenumbers.net/prptop/prptop.php

            If I know, both, an ouput of pfgw (without -tc option) and Henri's pages,
            contain (strong, but) PRP's not primes. Mark has written about primes.

            Wojtek


            ===============================================
            Wojciech Florek (WsF)
            Adam Mickiewicz University, Faculty of Physics
            ul. Umultowska 85, 61-614 Poznan, Poland

            Phone: (++48-61) 8295033 fax: (++48-61) 8257758
            email: Wojciech.Florek@...
          • Lelio sknet
            ... If so Mark is owing us his proof of (443,1774) Pfgw -f131071 -tc -q1775^^443-1774^^443 Primality testing 1775^443-1774^443 [N-1/N+1,
            Message 5 of 10 , Mar 5, 2006
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              > If I know, both, an ouput of pfgw (without -tc option) and Henri's pages,
              > contain (strong, but) PRP's not primes. Mark has written about primes.
              >
              > Wojtek

              If so Mark is owing us his proof of (443,1774)

              Pfgw -f131071 -tc -q1775^^443-1774^^443

              Primality testing 1775^443-1774^443 [N-1/N+1, Brillhart-Lehmer-Selfridge]
              Trial factoring to 228403237
              Running N-1 test using base 3
              Running N+1 test using discriminant 7, base 6+sqrt(7)
              Calling N-1 BLS with factored part 1.74% and helper 0.46% (5.73% proof)
              1775^443-1774^443 is Fermat and Lucas PRP! (32.1811s+0.0014s)
            • Mark Underwood
              My mistake, I should have said probable prime instead of prime. (I used GP Pari without the -tc option.) Mark ... Henri s pages, ... primes. ... Selfridge] ...
              Message 6 of 10 , Mar 5, 2006
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                My mistake, I should have said probable prime instead of prime.
                (I used GP Pari without the -tc option.)

                Mark


                --- In primenumbers@yahoogroups.com, "Lelio sknet" <lelio@...> wrote:
                >
                > > If I know, both, an ouput of pfgw (without -tc option) and
                Henri's pages,
                > > contain (strong, but) PRP's not primes. Mark has written about
                primes.
                > >
                > > Wojtek
                >
                > If so Mark is owing us his proof of (443,1774)
                >
                > Pfgw -f131071 -tc -q1775^^443-1774^^443
                >
                > Primality testing 1775^443-1774^443 [N-1/N+1, Brillhart-Lehmer-
                Selfridge]
                > Trial factoring to 228403237
                > Running N-1 test using base 3
                > Running N+1 test using discriminant 7, base 6+sqrt(7)
                > Calling N-1 BLS with factored part 1.74% and helper 0.46% (5.73%
                proof)
                > 1775^443-1774^443 is Fermat and Lucas PRP! (32.1811s+0.0014s)
                >
              • mikeoakes2@aol.com
                ... Mark You will find quite a lot of info in my NMBRTHRY post of 2001:- http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0105&L=NMBRTHRY&P=R359&I=-3 And, as
                Message 7 of 10 , Mar 12, 2006
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                  In an email dated Sun, 5 3 2006 3:56:53 am GMT, "Mark Underwood" <mark.underwood@...> writes:

                  >It is well known that for the expression
                  >
                  >(n+1)^x - n^x
                  >
                  >to yield a prime number, x must itself be prime.
                  >(And if x is a prime p the same expression contains only prime
                  >factors of the form p*2k + 1)
                  >
                  >Here's a question: For every prime p, is there always an n such that
                  >(n+1)^p - n^p is prime?
                  >
                  >I think so.  Here are the lowest 15 values of n which yield a prime
                  >for the first 15 primes (p,n):
                  >
                  >(2,1) (3,1) (5,1) (7,1) (11,5) (13,1) (17,1) (19,1) (23,5) (29,2)
                  >(31,1) (37,39) (41,6) (43,4) (47,12)
                  >
                  >(For example (12+1)^47 - 12^47 is prime. )
                  >
                  >Here is a list (p,n) which shows only increasing n:
                  >
                  >(2,1) (11,5) (37,39) (73,46) (137,327) (239,367) (307,402) (419,781)
                  >(443,1774)
                  >
                  >(For example (1774+1)^443 - 1774^443 is prime. (Over 1400 digits, my
                  >biggest ever found , wheee! :) )
                  >
                  >Another question is this: Given a prime p is there an upper bound for
                  >n before which we can be sure to generate a prime?
                  >The value (1 + 1/e) crops up alot in my dabblings, so for fun I'll
                  >guess the following:
                  >
                  >(n+1)^p - n^p is prime for some n < p^(1+1/e)
                  >
                  >Oh the rigour of it all (not!)...


                  Mark

                  You will find quite a lot of info in my NMBRTHRY post of 2001:-
                  http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0105&L=NMBRTHRY&P=R359&I=-3

                  And, as Lélio pointed out, Henri's
                  http://www.primenumbers.net/prptop/prptop.php
                  lists all known PRPs of this type with at least 10K digits.
                  You can use the search string "x^y-z^y" to find them - there are 468 of them.

                  A variant on one of your questions would be:-
                  Given n, is there an upper bound for p before which we can be sure to generate a prime?

                  The 2 cases n=48 and n=61 are tough, and resisted a solution until last year:-
                  27 48^58543-47^58543 98425 Jean-Louis Charton 11/2005
                  28 61^54517-60^54517 97331 Jean-Louis Charton 09/2005

                  My {n<=1000, p<=10000} project will be complete in a few more months, and I plan to make the results available on-line.

                  -Mike Oakes
                • Mark Underwood
                  Thanks Mike! Yes a very informative post. I did briefly investigate the variation on the theme which you mentioned, but I encountered the 48 problem and it
                  Message 8 of 10 , Mar 12, 2006
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                    Thanks Mike! Yes a very informative post. I did briefly investigate
                    the variation on the theme which you mentioned, but I encountered
                    the "48 problem" and it scared me away! I'm very glad to see someone
                    got a solution for it.

                    Can't wait to see the data from your project when it's finished. Who
                    knows, there might be an underlying principle lurking just waiting to
                    be uncovered.

                    Mark




                    --- In primenumbers@yahoogroups.com, mikeoakes2@... wrote:>

                    > You will find quite a lot of info in my NMBRTHRY post of 2001:-
                    > http://listserv.nodak.edu/cgi-bin/wa.exe?
                    A2=ind0105&L=NMBRTHRY&P=R359&I=-3
                    >
                    > And, as Lélio pointed out, Henri's
                    > http://www.primenumbers.net/prptop/prptop.php
                    > lists all known PRPs of this type with at least 10K digits.
                    > You can use the search string "x^y-z^y" to find them - there are
                    468 of them.
                    >
                    > A variant on one of your questions would be:-
                    > Given n, is there an upper bound for p before which we can be sure
                    to generate a prime?
                    >
                    > The 2 cases n=48 and n=61 are tough, and resisted a solution until
                    last year:-
                    > 27 48^58543-47^58543 98425 Jean-Louis Charton 11/2005
                    > 28 61^54517-60^54517 97331 Jean-Louis Charton 09/2005
                    >
                    > My {n<=1000, p<=10000} project will be complete in a few more
                    months, and I plan to make the results available on-line.
                    >
                    > -Mike Oakes
                    >
                  • Cletus Emmanuel
                    Mike, I just went to your site and you said I have just completed an investigation, starting 19 Aug 00, into primes of a form which does not seem to have
                    Message 9 of 10 , Mar 13, 2006
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                      Mike,
                      I just went to your site and you said

                      "I have just completed an investigation, starting 19 Aug 00, into primes of a
                      form which does not seem to have been studied in the past (correct me if I
                      am wrong):-"

                      Again I would like to note that these numbers were looked at back in (if i can recall- March of) 1995. I studied the numbers for my senior math seminar ( I was a senior Mathematics major) at the University of the Virgin Islands, St. Thomas, Virgin Islands. Back then I only looked at n from 2 to 9 and called them Mersenne Class Numbers. I went as far then (and I may be 100% wrong here) as saying that Mersenne and others knew of these numbers and their properties, but thought that 2^p - 1 was much more interesting in terms of finding and proving primes and therefore pursue that line. However, I found these numbers quite interesting then and I recon so did you and others who are now looking for large PRP of these forms.

                      To date, you've gone farther than I have in terms of the value of on n and the range of p. It is good to see that more and more people are finding these numbers interesting. I hope some day we can find a test to prove these numbers prime and that they can dominate the list of Largest Known Primes...


                      ---Cletus Emmanuel
                      In an email dated Sun, 5 3 2006 3:56:53 am GMT, "Mark Underwood" <mark.underwood@...> writes:

                      >It is well known that for the expression
                      >
                      >(n+1)^x - n^x
                      >
                      >to yield a prime number, x must itself be prime.
                      >(And if x is a prime p the same expression contains only prime
                      >factors of the form p*2k + 1)
                      >
                      >Here's a question: For every prime p, is there always an n such that
                      >(n+1)^p - n^p is prime?
                      >
                      >I think so. Here are the lowest 15 values of n which yield a prime
                      >for the first 15 primes (p,n):
                      >
                      >(2,1) (3,1) (5,1) (7,1) (11,5) (13,1) (17,1) (19,1) (23,5) (29,2)
                      >(31,1) (37,39) (41,6) (43,4) (47,12)
                      >
                      >(For example (12+1)^47 - 12^47 is prime. )
                      >
                      >Here is a list (p,n) which shows only increasing n:
                      >
                      >(2,1) (11,5) (37,39) (73,46) (137,327) (239,367) (307,402) (419,781)
                      >(443,1774)
                      >
                      >(For example (1774+1)^443 - 1774^443 is prime. (Over 1400 digits, my
                      >biggest ever found , wheee! :) )
                      >
                      >Another question is this: Given a prime p is there an upper bound for
                      >n before which we can be sure to generate a prime?
                      >The value (1 + 1/e) crops up alot in my dabblings, so for fun I'll
                      >guess the following:
                      >
                      >(n+1)^p - n^p is prime for some n < p^(1+1/e)
                      >
                      >Oh the rigour of it all (not!)...


                      Mark

                      You will find quite a lot of info in my NMBRTHRY post of 2001:-
                      http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0105&L=NMBRTHRY&P=R359&I=-3

                      And, as Lélio pointed out, Henri's
                      http://www.primenumbers.net/prptop/prptop.php
                      lists all known PRPs of this type with at least 10K digits.
                      You can use the search string "x^y-z^y" to find them - there are 468 of them.

                      A variant on one of your questions would be:-
                      Given n, is there an upper bound for p before which we can be sure to generate a prime?

                      The 2 cases n=48 and n=61 are tough, and resisted a solution until last year:-
                      27 48^58543-47^58543 98425 Jean-Louis Charton 11/2005
                      28 61^54517-60^54517 97331 Jean-Louis Charton 09/2005

                      My {n<=1000, p<=10000} project will be complete in a few more months, and I plan to make the results available on-line.

                      -Mike Oakes


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                      [Non-text portions of this message have been removed]
                    • grostoon
                      Hi all, ... last year:- ... I m currently working on 10^p-9^p and 138^p-137^p. 138^p-137^p appears to be harder than 48 and 61. Some time ago, I also started
                      Message 10 of 10 , Mar 22, 2006
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                        Hi all,

                        >
                        > The 2 cases n=48 and n=61 are tough, and resisted a solution until
                        last year:-
                        > 27 48^58543-47^58543 98425 Jean-Louis Charton 11/2005
                        > 28 61^54517-60^54517 97331 Jean-Louis Charton 09/2005
                        >

                        I'm currently working on 10^p-9^p and 138^p-137^p. 138^p-137^p
                        appears to be harder than 48 and 61.

                        Some time ago, I also started to build a table of the first n such
                        that (n+1)^p-n^p is prime or at least PRP. Here is the result :

                        p n n/p

                        2 1 0,50
                        3 1 0,33
                        5 1 0,20
                        7 1 0,14
                        11 5 0,45
                        13 1 0,07
                        17 1 0,05
                        19 1 0,05
                        23 5 0,21
                        29 2 0,06
                        31 1 0,03
                        37 39 1,05
                        41 6 0,14
                        43 4 0,09
                        47 12 0,25
                        53 2 0,03
                        59 2 0,03
                        61 1 0,01
                        67 6 0,08
                        71 17 0,23
                        73 46 0,63
                        79 7 0,08
                        83 5 0,06
                        89 1 0,01
                        97 25 0,25
                        101 2 0,01
                        103 41 0,39
                        107 1 0.00
                        109 12 0,11
                        113 7 0,06
                        127 1 0.00
                        131 7 0,05
                        137 327 2,38
                        139 7 0,05
                        149 8 0,05
                        151 44 0,29
                        157 26 0,16
                        163 12 0,07
                        167 75 0,44
                        173 14 0,08
                        179 51 0,28
                        181 110 0,60
                        191 4 0,02
                        193 14 0,07
                        197 49 0,24
                        199 286 1,43
                        211 15 0,07
                        223 4 0,01
                        227 39 0,17
                        229 22 0,09
                        233 109 0,46
                        239 367 1,53
                        241 22 0,09
                        251 67 0,26
                        257 27 0,10
                        263 95 0,36
                        269 80 0,29
                        271 149 0,54
                        277 2 0.00
                        281 142 0,50
                        283 3 0,01
                        293 11 0,03
                        307 402 1,30
                        311 3 0.00
                        313 44 0,14
                        317 10 0,03
                        331 82 0,24
                        337 20 0,05
                        347 95 0,27
                        349 4 0,01
                        353 108 0,30
                        359 349 0,97
                        367 127 0,34
                        373 303 0,81
                        379 37 0,09
                        383 3 0.00
                        389 162 0,41
                        397 119 0,29
                        401 8 0,01
                        409 128 0,31
                        419 781 1,86
                        421 5 0,01
                        431 25 0,05
                        433 75 0,17
                        439 189 0,43
                        443 1774 4,00
                        449 197 0,43
                        457 360 0,78
                        461 13 0,02
                        463 128 0,27
                        467 155 0,33
                        479 20 0,04
                        487 210 0,43
                        491 1550 3,15
                        499 3 0.00
                        503 1105 2,19
                        509 389 0,76
                        521 1 0.00
                        523 16 0,03
                        541 3 0.00
                        547 32 0,05
                        557 353 0,63
                        563 4 0.00
                        569 164 0,28
                        571 16 0,02
                        577 207 0,35
                        587 154 0,26
                        593 2175 3,66
                        599 3 0.00
                        601 604 1,00
                        607 1 0.00
                        613 104 0,16
                        617 348 0,56
                        619 251 0,40
                        631 128 0,20
                        641 1469 2,29
                        643 12 0,01
                        647 2 0.00
                        653 1167 1,78
                        659 13 0,01
                        661 100 0,15
                        673 22 0,03
                        677 193 0,28
                        683 1147 1,67
                        691 46 0,06
                        701 54 0,07
                        709 4 0.00
                        719 252 0,35
                        727 209 0,28
                        733 822 1,12
                        739 165 0,22
                        743 4 0.00
                        751 429 0,57
                        757 572 0,75
                        761 1973 2,59
                        769 56 0,07
                        773 27 0,03
                        787 81 0,10
                        797 33 0,04
                        809 509 0,62
                        811 73 0,09
                        821 180 0,21
                        823 94 0,11
                        827 91 0,11
                        829 101 0,12
                        839 359 0,42
                        853 29 0,03
                        857 81 0,09
                        859 155 0,18
                        863 629 0,72
                        877 21 0,02
                        881 666 0,75
                        883 154 0,17
                        887 329 0,37
                        907 172 0,18
                        911 722 0,79
                        919 943 1,02
                        929 264 0,28
                        937 459 0,48
                        941 989 1,05
                        947 699 0,73
                        953 968 1,01
                        967 1442 1,49
                        971 950 0,97
                        977 163 0,16
                        983 353 0,35
                        991 653 0,65
                        997 88 0,08
                        1009 553 0,54
                        1013 99 0,09
                        1019 3579 3,51
                        1021 889 0,87
                        1031 1654 1,60
                        1033 923 0,89
                        1039 5 0.00
                        1049 2741 2,61
                        1051 40 0,03
                        1061 2 0.00
                        1063 274 0,25
                        1069 339 0,31
                        1087 45 0,04
                        1091 72 0,06
                        1093 416 0,38
                        1097 1733 1,57
                        1103 347 0,31
                        1109 80 0,07
                        1117 90 0,08
                        1123 58 0,05
                        1129 10 0.00
                        1151 1094 0,95
                        1153 488 0,42
                        1163 39 0,03
                        1171 135 0,11
                        1181 33 0,02
                        1187 50 0,04
                        1193 3 0.00
                        1201 290 0,24
                        1213 1315 1,08
                        1217 473 0,38
                        1223 2790 2,28
                        1229 2122 1,72
                        1231 349 0,28
                        1237 414 0,33
                        1249 593 0,47
                        1259 424 0,33
                        1277 44 0,03
                        1279 1 0.00
                        1283 2058 1,60
                        1289 5143 3,98
                        1291 384 0,29
                        1297 871 0,67
                        1301 69 0,05
                        1303 55 0,04
                        1307 96 0,07
                        1319 490 0,37
                        1321 630 0,47
                        1327 6 0.00
                        1361 874 0,64

                        In fact, it seems that most of the time the first n is small and even
                        <= p but there are a few counterexamples.
                        My actual record is for p=443 for which n = 1774 (n/p = 4.0045).


                        I believe that the 2 following assertions are both true even if (of
                        course) they seems currently impossible to demonstrate:

                        - For every prime p there exist an integer n such that (n+1)^p-n^p is
                        prime (1)

                        - For every integer n >= 1 there exist a prime p such that (n+1)^p-
                        n^p is prime (2)


                        And more than that, I believe that the sets of such n (1) and such p
                        (2) are infinite


                        J-L
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