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A prime rich sequence???

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  • Kermit Rose
    3 = 2 + 1 5 = 2^2 + 1 7 = 2^2 + 2 + 1 11 = 2^3 + 2 + 1 13 = 2^3 + 2^2 + 1 17 = 2^4 + 1 3 + 1 = 2 * 2 3^2 + 1 = 2 * 5 3^2 + 3 + 1 = 13 = prime 3^3 + 3 + 1 = 31
    Message 1 of 2 , Nov 29, 2006
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      3 = 2 + 1
      5 = 2^2 + 1
      7 = 2^2 + 2 + 1
      11 = 2^3 + 2 + 1
      13 = 2^3 + 2^2 + 1
      17 = 2^4 + 1

      3 + 1 = 2 * 2
      3^2 + 1 = 2 * 5
      3^2 + 3 + 1 = 13 = prime
      3^3 + 3 + 1 = 31 = prime
      3^3 + 3^2 + 1 = 37 = prime
      3^4 + 1 = 2 * 41

      4 + 1 = 5 = prime
      4^2 + 1 = 17 = prime
      4^2 + 4 + 1 = 3 * 7 ; 2^2 + 2 + 1 = 7
      4^3 + 4 + 1 = 3 * 23 ; 2^3 + 2 + 1 is neither 23 nor 3.
      4^3 + 4^2 + 1 = 3^4
      4^4 + 1 = 257 = prime

      5+1 = 2 * 3
      5^2 + 1 = 2 * 13
      5^2 + 5 + 1 = 31 = prime
      5^3 + 5 + 1 = 131 = prime
      5^3 + 5^2 + 1 = 151 = prime

      6 + 1 = 7 = prime
      6^2 + 1 = 37 = prime
      6^2 + 6 + 1 = 43 = prime
      6^3 + 6 + 1 = 223 = prime
      6^3 + 6^2 + 1 = 11 * 23 ; 2^3 + 2^2 + 1 = 11

      Does the prime richness extend to larger numbers, or is this an illusion
      of small primes?

      Kermit < kermit@... >
    • Joshua Zucker
      I don t think these are particularly good prime-generating polynomials you ve found. Perhaps they are quite good given the smallness of the coefficients,
      Message 2 of 2 , Nov 30, 2006
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        I don't think these are particularly good prime-generating polynomials
        you've found. Perhaps they are quite good given the smallness of the
        coefficients, though! Most of the examples below have much larger
        coefficients than just 0 and 1.

        For some really excellent ones, take a look at
        http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
        http://www.maa.org/editorial/mathgames/mathgames_07_17_06.html
        or toward the bottom of
        http://euler.free.fr/contest/PGPReport.htm
        where they give polynomials both with integer and rational coefficients,
        and which give lots of consecutive primes,
        or which give a relatively high proportion of primes in some range
        like x = 0 to 1000 ...

        --Joshua Zucker
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