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novel primality principle pondered

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  • aldrich617
    Actually, I was looking for a decidedly non-Fibonacci way of ordering an infinite series of valid fourth degree +1 mod 10 sequences. Embedded in the third and
    Message 1 of 1 , May 26, 2006
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      Actually, I was looking for a decidedly non-Fibonacci way of
      ordering an infinite series of valid fourth degree +1 mod 10
      sequences.

      Embedded in the third and fourth instances of +/- 1 mod 10
      quadratic sequences associated with the primes described below
      and indeed for all primes +1 mod 10, there appears to one of
      these fourth degree sequences that starts with the quadratic's
      term numbers 1,2,5,10,17.
      (For 881 in the third instance: 881^2 782611 802021 834571
      880561; For 881 in the fourth instance: 881^2 788551 825781
      888031 975601) If this is true, it may be the basis for a valid
      rule for a different variety of infinite series.

      Certain other invariance in the patterns below then suggested an
      interesting primality principle, and several related conjectures.
      There definitely does seem to be an unexpected connection between
      the occurrence or non-occurrence of +/- 1 mod 10 quadratic
      sequences that start with a (test#)^2 and :

      (1) a conceptually simple (but difficult to implement) way to
      determine if the test# is composite or prime.
      (2) the presence or absence of factors ending in 3 or 7 in the
      test# (also difficult to implement).

      Test numbers are limited to integers ending in 1 or 9.

      Begin with the square of an integer (T) to be tested for primality.
      If the first increment (I) of the fourth +/- mod 10 quadratic
      sequence emanating from this square is known then:

      If I/10 > T , then T is prime,
      Else If I/10 < T, then T is composite

      The test#'s listed below are followed by the first increment/10
      in their first four instances where valid +/- 1 mod 10 quadratic
      sequences are found :

      881^2 : 397, 441, 645, 1239
      3001^2: 1489, 1501, 2953, 3051
      3251^2: 1626, 1639, 3201, 3303

      In each instance, the fourth number is higher than the test# (T),
      which is prime:
      (881 < 1239), (3001 < 3051), (3251 < 3303)
      I have been unable to find any counterexamples to this type of
      pattern.

      I have also been unable to find any counterexamples for the pattern
      for composites:

      3781: (19*199)^2 : 1692, 1891, 1905, 1923
      5611: (31*181)^2: 2533, 2625, 2806, 2823

      In each instance, the fourth number is lower than the test # (T) ..
      Therefore, if the pattern holds up, establishing that an integer
      ending in one is composite is again a matter of checking the
      fourth instance of the (first increment)/10. If the fourth
      occurrence is < T then the test number is composite:
      (3781 > 1923), (5611 > 2823).

      The same principle of primality seems to apply to test#'s ending
      in 9.

      229^2: 103, 115, 159, 345
      209: (19*11)^2 : 94, 99, 105, 117


      To illustrate (2) above, let us start with the square of the
      prime test# 881 , then calculate the first increment from
      that to the next member of the sequence that we find gratis
      for every test# at 10*(T + 1)/2. (For 881,this is the second
      instance a valid +/-mod 10 quadratic)

      Thus 10*(881 +1)/2 = 4410. Next add: 881 + 4410 = 5291. This
      is the second term of the quadratic sequence. Then add the
      constant increment 10 to the first increment to get 4420,
      and use this to get 9711, the third term of the quadratic
      sequence. Finally we get the sequence 881, 5291, 9711 ….. ,
      the first three terms of a valid +/-mod 10 quadratic.

      If the test# had been 901 = 17*53, then the first increment
      of the quadratic, 4510,would not start a valid +/- mod 10
      quadratic sequence. There are factors of 17 and 53 in this
      sequence.

      In examing these patterns, it is evident certain other
      conjectures concerning primality and the location of valid
      +/- mod 10 quadratic sequences relating to primes and composites
      are likely true:

      1) There is always a valid sequence (G) beginning with a first
      increment of 10*(test# +1)/2 for test numbers with factors
      ending exclusively in 1 or 9,and never one for test numbers with
      factors ending in 3 or 7.
      2) There is floor value (F) at approximately .4473 * T, below
      which no (first increment)/10 can occur: For example, for 229
      the floor value is 102.
      3) If two (sometimes one) valid quadratics can be found between
      the floor value and the gratis quadratic at (test# +1)/2, a
      test# can be shown to be composite. For 209 (listed above),
      showing that 93 (F) < 94 I/10) and 99 (I/10) < 105 (G) would
      be sufficient.

      As of the moment, none of the preceding conjectures has been
      proven, though I believe counterexamples are unlikely, and if
      they are proven, they may not be usable. For those who enjoy
      contemplating the nature of primality this does not really
      matter because the mysterious thing-in-itself gives gratifying
      flashes of insight. Practical application ,however, must await a
      better means of calculating the location of all the instances
      of +/- mod 10 quadratics (if there is one).
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