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[PrimeNumbers] Re: A small observation

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  • mikeoakes2@aol.com
    Robert Smith wrote to primenumbers@yahoogroups.com ... Expanding by the binomial theorem, (p+1)q-1 is divisible by p. If you make use
    Message 1 of 3 , Dec 23, 2006
      Robert Smith <rw.smith@...> wrote to
      primenumbers@yahoogroups.com

      >For prime p, there are no prime factors of (p+1)^q-1 smaller than p
      >for all q<p, q prime.
      >

      Expanding by the binomial theorem, (p+1)q-1 is divisible by p.
      If you make use of the fact that all other prime factors of (p+1)^q-1
      are of the form
      f = k*q+1,
      where k must be even if f is odd,
      you should readily be able to prove why your observed patterns occur.

      Mike


      -----Original Message-----
      From: rw.smith@...
      To: primenumbers@yahoogroups.com
      Sent: Sat, 23 Dec 2006 8.12AM
      Subject: [PrimeNumbers] Re: A small observation

      A further bit of research shows that for primes q up to 127, certain
      values of q (17, 37) do not have prime factors smaller than p for all
      p+1 in (p+1)^q-1

      The factors of (p+1)^q-1 is given for q=17 for all p to p=127 below.
      The smallest factor in each case is p:

      p+1 factors of (p+1)^q-1, q=17
      3 2*1871*34511
      4 3*43691*131071
      6 5*239*409*1123*30839
      8 7*103*2143*11119*131071
      12 11*2693651*74876782031
      14 13*103*22771730193675277
      18 17^2*7563707819165039903
      20 19*689852631578947368421
      24 23*307*120574031*341563234253
      30 29*103*409*10570676926829627653
      32 31*131071*9520972806333758431
      38 37*137*4269211*33193361176106273
      42 41*1158007*82935963690420160273
      44 43*14519*13908821686834391538539
      48 47*C_810966991148301450330945841
      54 53*11270423*C_472582712990569339157
      60 59*137*57223*3659553116774508114611
      62 61*48453916488902607769120106731
      68 67*1973*309469*1240559*280036765431547
      72 71*103*137*114320887*327876819251626913
      74 73*137*258469*23146607407263591330467
      80 79*152729*4885801*3819842050386190289
      84 83*6218272796370530483675222621221
      90 89*71041879*C_263765636099232967211329
      98 97*2687*11527*4530671*521104791402957529
      102 101*307*16661*87313*310429650430900811021
      104 103*C_189116552966629126613043005678761
      108 107*14281*17783*C_1361621555332298324123027
      110 109*6121*12377*4221271*C_1450000477088161373
      114 113*307*38183*C_70031851163818972855046051
      128 127*239*20231*131071*131105292137*62983048367

      Contrast with q=3, where p is only sometimes the smallest factor:

      p+1 factors of (p+1)^q-1, q=3
      3 2*13
      4 3^2*7
      6 5*43
      8 7*73
      12 11*157
      14 13*211
      18 7^3*17
      20 19*421
      24 23*601
      30 7^2*19*29
      32 7*31*151
      38 37*1483
      42 13*41*139
      44 7*43*283
      48 13*47*181
      54 53*2971
      60 7*59*523
      62 61*3907
      68 13*19^2*67
      72 7*71*751
      74 7*13*61*73
      80 79*6481
      84 37*83*193
      90 89*8191
      98 31*97*313
      102 7*19*79*101
      104 67*103*163
      108 61*107*193
      110 109*12211
      114 7*113*1873
      128 7^2*127*337

      However I have a feeling that this may be a case of "sods law of small
      numbers". Trouble is I have no maths software to test these results
      further.

      Regards

      Robert Smith
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