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17867Re: Cubic x^3 + x^2 + x + t prime generators

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  • Mark Underwood
    Mar 28, 2006
      --- In primenumbers@yahoogroups.com, "Kermit Rose" <kermit@...> wrote:
      >
      >
      > In order that x^3 + x^2 + x + t not have any positive prime factors
      < 43,
      >
      > t must be
      >
      > 2 mod 3
      >
      > and
      >
      > 3 or 4 mod 5
      >
      > and
      >
      > 2 or 5 mod 7
      >
      > and
      >
      > 2 or 3 or 7 or 9 mod 11
      >
      > and

      > (snip)


      Kermit, what is funny is that I overlooked the obvious:
      x^3+x^2+x+k contains a factor of two for every other x! Brought down
      by the lowly factor of two.

      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
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