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

generating primes with p^n + 2^m

Expand Messages
  • Mark Underwood
    Hi all I ve been toying with the expression p^n + 2^m as a prime generator, where p is a prime and n and m are from 1 to p. Based on the little data I have
    Message 1 of 1 , Jul 2, 2003
    • 0 Attachment
      Hi all

      I've been toying with the expression p^n + 2^m as a prime generator,
      where p is a prime and n and m are from 1 to p.

      Based on the little data I have looked at (below), I would hazard a
      guess that for every odd p, p^n + 2^m produces at least one prime.
      Perhaps at least 8 primes. (With more data I would like to put
      tighter restraints on that guess.) Also, I would guess that over the
      range of odd p's, p^n + 2^m will generate any and every prime at
      least once.

      Here is the data. After this it gets a little more interesting, so
      bear with this. Also, since I did this manually there may be an error
      somewhere.

      3^1 + 2^1 = 5
      3^1 + 2^2 = 7
      3^1 + 2^3 = 11
      3^2 + 2^1 = 11
      3^2 + 2^2 = 13
      3^2 + 2^3 = 17
      3^3 + 2^1 = 29
      3^3 + 2^2 = 31
      (8 primes generated)


      5^1 + 2^1 = 7
      5^1 + 2^3 = 13
      5^1 + 2^5 = 37
      5^2 + 2^2 = 29
      5^2 + 2^4 = 41
      5^3 + 2^1 = 127
      5^3 + 2^5 = 157
      5^4 + 2^4 = 641
      (8 primes generated)

      7^1 + 2^2 = 11
      7^1 + 2^4 = 23
      7^1 + 2^6 = 71
      7^2 + 2^2 = 53
      7^2 + 2^6 = 113
      7^3 + 2^2 = 347
      7^3 + 2^4 = 359
      7^4 + 2^4 = 2417
      7^5 + 2^2 = 16811
      7^5 + 2^4 = 16823
      7^5 + 2^6 = 16871
      7^7 + 2^2 = 823547
      (12 primes generated)

      11^1 + 2^1 = 13
      11^1 + 2^3 = 19
      11^1 + 2^5 = 43
      11^1 + 2^7 = 139
      11^1 + 2^9 = 523
      11^2 + 2^4 = 137
      11^3 + 2^7 = 1459
      11^4 + 2^4 = 14657
      11^4 + 2^8 = 14897
      11^5 + 2^1 = 161053
      11^5 + 2^3 = 161059
      11^5 + 2^9 = 161563
      11^7 + 2^9 = 19487683
      11^7 + 2^11 = 19489219
      11^9 + 2^1 = 2357947693
      11^9 + 2^5 = 2357947723
      (16 primes generated)

      13^1 + 2^2 = 17
      13^1 + 2^4 = 29
      13^1 + 2^6 = 79
      13^1 + 2^8 = 269
      13^2 + 2^2 = 173
      13^2 + 2^6 = 233
      13^2 + 2^10 = 1193
      13^3 + 2^4 = 2213
      13^3 + 2^10 = 3221
      13^5 + 2^8 = 371549
      13^6 + 2^2 = 4826813
      13^8 + 2^8 = 815730977
      13^10 + 2^2 = 137858491853
      13^13 + 2^4 = 302875106592269
      (14 primes generated)


      Surely it is just a coinicidence that in all 5 cases the number of
      primes generated is even?

      Now, if we restrict the exponents m and n so that each are of the
      form 2^c, (c can be zero) we get

      3^1 + 2^1 = 5
      3^1 + 2^2 = 7
      3^2 + 2^1 = 11
      3^2 + 2^2 = 13
      (4 primes generated)

      5^1 + 2^1 = 7
      5^2 + 2^2 = 29
      5^2 + 2^4 = 41
      5^4 + 2^4 = 641
      (4 primes generated)

      7^1 + 2^2 = 11
      7^1 + 2^4 = 23
      7^2 + 2^2 = 53
      7^4 + 2^4 = 2417
      (4 primes generated)

      11^1 + 2^1 = 13
      11^2 + 2^4 = 137
      11^4 + 2^4 = 14657
      11^4 + 2^8 = 14897
      (4 primes generated)

      13^1 + 2^2 = 17
      13^1 + 2^4 = 29
      13^1 + 2^8 = 269
      13^2 + 2^2 = 173
      13^8 + 2^8 = 815730977
      (5 primes generated)

      Now I add a few more primes, but my computer's calculator only goes
      so high and then looses digit precision, so each is incomplete.


      17^1 + 2^1 = 19
      17^2 + 2^2 = 293
      17^4 + 2^4 = 83537
      17^4 + 2^8 = 83777
      17^4 + 2^16 = 149057
      17^8 + 2^4 = 6975757457
      17^8 + 2^16 = 6975822977
      17^16 ???

      19^1 + 2^2 = 23
      19^2 + 2^8 = 617
      19^4 + 2^4 = 130337
      19^8 + 2^16 = 16983628577
      19^16 ???

      23^4 + 2^4 = 279857
      23^4 + 2^8 = 280097
      23^16 ???

      29^1 + 2^1 = 31
      29^2 + 2^8 = 1097
      29^2 + 2^16 = 66377
      29^8 ???
      29^16???

      It is tempting to further guess that there is at least one prime
      generated for each of these. But what I found very surprising was
      this: if the two exponents add up to an even number greater than 7
      then the generated prime seems to always ends in a seven! I haven't
      yet investigated why this might be. Maybe it's just a temporary blip
      but it is interesting.


      Mark
    Your message has been successfully submitted and would be delivered to recipients shortly.