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

prime generating polynomials

Expand Messages
  • Kermit Rose
    Mark Underwood wrote: However here are some things to chew on: As we know, x^2 + x + 41 contains no prime factors below 41. But
    Message 1 of 2 , Mar 29, 2006
    • 0 Attachment
      Mark Underwood <mark.underwood@...> wrote:

      However here are some things to chew on:

      As we know, x^2 + x + 41 contains no prime factors below 41.

      But who knew that -x^10 + x^2 + 43 contains no prime factors below
      43? And in a strange twist, x^10 - x^2 + 43 also contains no prime
      factors below 43, even though their factor set appears to be
      different. x^10 - x^2 + 43 has as its lowest factors 43,73,89,107 and
      113.

      Even more fun, x^8 + x^4 + 59 has no prime factors below 59.

      And this is interesting:

      x^8 - x^6 - x^4 -x^2 + 1 has no prime factors under 43.

      And to top it all off:

      x^8 - x^4 + 1 has no prime factors below 73 (!)

      Even more incredible is that its lowest five prime factors are
      73,97,193,241,337. This is such a glaring scarcity of low prime
      factors. I can see that the primes must be of the form 6n+1, but
      these appear to go beyond that, to the form 24n+1.

      Alot to investigate here, such as the existence of similar forms
      which are even poorer in prime factors.

      Mark


      Kermit says:

      Examine

      x^10 - x^2 + 43

      mod 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41


      First examine x^10 - x^2

      x^10 - x^2 = x^2 [ x^8 - 1 ] = x^2 [ x^4 + 1] [ x^2 + 1 ] [x + 1] [ x
      - 1]

      x^10 - x^2 is always zero mod 2 since x(x-1) = 0 mod 2

      x^10 - x^2 is always zero mod 3 since x(x-1)(x+1) = 0 mod 3

      x^10 - x^2 is always zero mod 5 since x(x4-1) = 0 mod 5

      Mod 7
      x^2 : 0
      x-1: 1
      x+1: 6
      x^10 - x^2 is zero mod 7, for x = 0, or x=1 or x = 6

      2^10 - 2^2 mod 7 = 5
      3^10 - 3^2 mod 7 = 2
      4^10 - 4^2 mod 7 = 2
      since 3 + 4 = 7
      5^10 - 5^2 = 5
      since 2 + 5 = 7.




      x^8 + x^4 + 59 mod 2, 3, 5, etc

      x^8 + x^4 = x^4 ( x^4 + 1)


      is always 0 mod 2
      is never = 1 mod 3
      is never = 1 or 3 or 4 mod 5
      etc


      x^8 - x^4 + 1 has no prime factors below 73 (!)


      (x^4)^2 - x^4 + 1 mod 2 = 1

      (x^4)^2 - x^4 + 1 mod 3 = 1

      (x^4)^2 - x^4 + 1 mod 5 = 1

      (x^4)^2 - x^4 + 1 mod 7 = 1 or 3 or 6

      etc


      > Even more incredible is that its lowest five prime factors are
      73,97,193,241,337. This is such a glaring scarcity of low prime
      factors. I can see that the primes must be of the form 6n+1, but
      these appear to go beyond that, to the form 24n+1.


      x^8 - x^4 + 1 is equal to 1 in mod 2, mod 3, mod 4, mod 5, mod 8, mod 16,

      and therefore

      x^8 - x^4 + 1 is equal to 1 mod 240

      Of course this says nothing about what the lowerest prime factors would be.
    Your message has been successfully submitted and would be delivered to recipients shortly.