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Primes and Composites occurring in Certain Equations

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  • aldrich617
    Conjectures concerning Primes and Composites of the Form 20x^2 -1, 20x^2 + 20x + 1, 20x^2 + 20x - 11, 20x^2 + 60x - 19, 20x^2 +180x +149 and 20x^2 + 340x +
    Message 1 of 1 , Dec 21, 2008
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      Conjectures concerning Primes and Composites of the Form
      20x^2 -1, 20x^2 + 20x + 1, 20x^2 + 20x - 11,
      20x^2 + 60x - 19, 20x^2 +180x +149 and 20x^2 + 340x + 421:.

      These equations are the first six members of a family of
      forms derived originally from 5x^2 + 5x +1
      (from the puzzle of 9/8/8), and they share some of its
      special properties. Prime and composite terms may be
      distinguished by simply counting the number of squares
      ocurring in in a relatively small calculated interval. Although
      this assertion remains to be proven, it seems a good bet,
      as the first 10000 iterations of each sequence yielded no
      counterexamples. All primes composing terms of each of
      these six equations are always +/- 1 mod 10 , and the
      constant second difference between three consecutive
      terms is always 40.

      The first two conjectures are a recasting of 5) and 6)
      from the puzzle of 12/8/8 (still underway). This recasting
      will now include in the count of squares the two non-
      functional squares occuring just outside of the interval
      that we examined then (old interval + 2) .In all cases a
      count of squares equal to two indicates a term is prime,
      and three or more that it is composite. This way of
      knowing extends to all six equations, and will also extend
      to additional members of the family (infinite?) of forms
      that can be derived.

      For All Conjectures below:
      x, A(x), k, T(k) : integers;
      T(k) = 5*(2*k -1)^2 - 4*A(x) ;

      Let A(x) = 20x^2 - 1;
      5a) A(x) will be prime if exactly two k values
      exist in the interval 2x < k < 3x + 2 such that T(k) is a
      square of an integer.

      Example: if x = 5, then A(x) = 499, 10< k < 17
      and the values of T(k) are:
      209, 649, 1129, 1649 , 47^2, 53^2.
      Two of these are squares, therefore 499 is prime.

      Let A(x) = 20x^2 - 1;
      6a) A(x) will be composite if there exists at
      least three k values in the interval 2x < k < 3x + 2
      such that T(k) is a square of an integer.

      Example: if x = 4, then A(x) = 319, 8< k < 14
      and the values of T(k) are:
      13^2, 23^2, 929 ,37^2, 43^2.
      Four of these are squares, therefore 319 is composite.

      Let A(x) = 20x^2 + 20x + 1;
      9) A(x) will be prime if exactly two k values
      exist in the interval 2x +1 < k < 3x + 4 such
      that T(k) is a square of an integer.

      Example: if x = 3, then A(x) = 241, 7< k < 13
      and the values of T(k) are:
      161, 481, 29^2, 1241, 41^2
      Two of these are squares, therefore 241 is prime.

      Let A(x) = 20x^2 + 20x + 1;
      10) A(x) will be composite if there exists at
      least three k values in the interval 2x + 1< k < 3x + 4
      such that T(k) is a square of an integer.

      Example: if x = 2, then A(x) = 121, 5< k < 10
      and the values of T(k) are:
      11^, 19^2 ,641, 31^2.
      Three of these are squares, therefore 121 is composite.

      Let A(x) = 20x^2 + 20x - 11;
      11) A(x) will be prime if exactly two k values
      exist in the interval 2x + 1 < k < 3x + 5 such
      that T(k) is a square of an integer.

      Example: if x = 4, then A(x) = 389, 9< k < 17
      and the values of T(k) are:
      249, 649, 33^2, 1569, 2089, 2649, 57^2
      Two of these are squares, therefore 389 is prime.

      Let A(x) = 20x^2 + 20x - 11;
      12) A(x) will be composite if there exists at
      least three k values in the interval 2x + 1< k < 3x + 5
      such that T(k) is a square of an integer.

      Example: if x = 5, then A(x) = 589, 11< k < 20
      and the values of T(k) are:
      17^2, 769, 1289, 43^2, 2449, 3089, 3769, 67^2
      Three of these are squares, therefore 589 is composite

      Let A(x) = 20x^2 + 60x - 19;
      13) A(x) will be prime if exactly two k values
      exist in the interval 2x +3 < k < 3x + 10
      such that T(k) is a square of an integer.

      (*note: from this point on only the square values
      of T(k), and the first value of T(k) will appear in
      the examples. All other T(k) will be designated as '-'.

      Example: if x = 4, then A(x) = 541, 11< k < 22 ,
      and the values of T(k) are:
      481 31^2 - - - - - - - 79^2
      Two of these are squares, therefore 541 is prime.

      Let A(x) = 20x^2 + 60x - 19;
      14) A(x) will be composite if there exists at
      least three k values in the interval 2x +3< k < 3x + 10
      such that T(k) is a square of an integer.

      Example: if x = 3, then A(x) = 341, 9< k < 19
      and the values of T(k) are:
      21^2 29^2 - - - - - - 69^2
      Three of these are squares, therefore 341 is composite.

      Let A(x) = 20x^2 +180x +149;
      15) A(x) will be prime if exactly two k values
      exist in the interval 2x + 9< k < 3x + 23
      such that T(k) is a square of an integer.

      Example: if x = 1, then A(x) = 349, 11< k < 26
      and the values of T(k) are:
      1249 -- 53^2 - - - - - - - - - 103^2
      Two of these are squares, therefore 349 is prime.

      Let A(x) = 20x^2 +180x +149;
      16) A(x) will be composite if there exists at
      least three k values in the interval
      2x + 9< k < 3x + 23 such that T(k) is a
      square of an integer.

      Example: if x = 2, then A(x) = 589, 13< k < 29
      and the values of T(k) are:
      1289 43^2 - - - 67^2 - - 83^2 - - - - - 113^2
      Four of these are squares, therefore 589 is composite.

      Let A(x) = 20x^2 + 340x + 421;
      17) A(x) will be prime if exactly two k values
      exist in the interval 2x + 17< k < 3x + 43
      such that T(k) is a square of an integer.

      Example: if x = 5, then A(x) = 2621, 27< k < 58
      and the values of T(k) are:
      4641 - - - - - - - - - - - - - - - - - - 181^2
      - - - - - - - - - 231^2
      Two of these are squares, therefore 2621 is prime.

      Let A(x) = 20x^2 + 340x + 421;
      18) A(x) will be composite if there exists at
      least three k values in the interval
      2x + 17 < k < 3x + 43 such that T(k) is a
      square of an integer.

      Example: if x = 4, then A(x) = 2101, 25< k < 55
      and the values of T(k) are:
      4601 - - - - 101^2 - - - - - - - -151^2
      - - - - - - - - - - - - - 221^2
      Three of these are squares, therefore 2101 is composite.

      Aldrich Stevens
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