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Re: Primes and Squares1 - Clarifications2

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  • richard042@yahoo.com
    The best way to avoid torturing a mathematician is to say nothing, or maybe talk about the weather. Barring that, clarity and conciseness expressed in terms
    Message 1 of 2 , Jan 2, 2004
      The best way to avoid torturing a mathematician is to say nothing, or
      maybe talk about the weather. Barring that, clarity and conciseness
      expressed in terms familiar to them is the only medicine that eases
      the pain. Remember, short does not guarantee concise. Even us
      amateur geniuses aren't smart enough to always know (or even
      discover) the correct mathematical term to use in a given instance.
      But, even though we might not have the correct terminology at our
      disposal, making up new terminology only makes matters worse and
      should be avoided if possible. The statement "odd divisors not
      exceeding n" would be understood, whereas a new term "primary
      divisors" requires a definition and increases the
      reader's "conceptual burden" <--Do you know what that term means?,
      neither do I.

      The mirror sequence you speak of is well known and plays a role in
      the foundations of many theorems and allows prime hunters to save
      tons of computer cycles, which must be a lot since they don't have

      Suggested reading - Wilson's Theorem, Quadratic Reciprocity,
      LeGendre's Prime Counting function, the theories of linear and
      quadratic congruences, theory of quadratic residues...

      Keywords - primorial, factorial, sigma divisor function, Euler
      Totient function, Jacobi symbol...

      ...anybody else have suggestions?

      > MIRROR SEQUENCE clarification
      > As I stated in my original submittal, the mirror sequence is a
      > sequence of primary odd divisors that is centered at 2*the product
      > all odd integers to N and spans +/- N from that center.

      The divisor structure you describe centered on ...20789, 20791... ,
      is exactly the same as that of ...-1, 1... , there is no need to look
      at such a large number only for this information. That number, by
      the way, = n!! for odd n, where !! is the double factorial function.

      The density distribution of all odd primes <= sqrt(n) as relates to
      their occurence as a factor in a sequence of n consecutive odd
      integers, is neither maximum nor a minimum when centered over {-
      n,...,n} and cannot be used as a bound.

      Here's an example
      For, {1,2,3,4,5,6,7,8,9}, 3 divides elements {3,6,9}, 3 out 9
      elements, density=1/3
      For, {1,3,5,7,9,11,13,15}, same result, 3 divides 3 out of 9 elements
      For, {15,17,19,21,23,25,27}, 3 divides 3 out of 7 elements > 1/3
      For, {1,3,5,7,9,11,13,15,17,19}, 3 divides 3 out of 10 elements < 1/3

      The pattern within n odd numbers centered on n!! is the exact same
      pattern as the n odd numbers centered on zero. All such patterns for
      any given span of n consecutive odd numbers, transposed say, a
      distance x from the center point at 0, will also occur as a mirror
      image at n!!-x, this is true. In fact, all unique, possible
      combinations are exhausted as the center point moves from 0 to 2 to
      4...to (n!!-2) to n!! Despite being "equivalent", the mirror image
      pairs are unique from each other within our frame of reference of
      increasing odd integers. Except that the sequences centered on 0 and
      n!! are not only equivalent, the are equal and identical. The
      sequence centered on 3 is identical to that centered on n!! + 3,
      while it's mirror image occurs at n!!-3 and repeats every k*n!!
      transposition of it's center, k = -inf, +inf. Thus there are n!!
      unique divisor distribution patterns for a sequence of n consecutive
      odd integers when only the odd divisors not exceeding n are
      considered in the make-up of the pattern. If mirror images are not
      considered unique, there are (n!!+1)/2 unique distributions.

      > Now lets match up the REF set with it's divisors from the LE
      > set.
      > REF set
      > 20779 20781 20783 20785 20787 20789 20791 20793 20795
      > 20799 20801
      > LE set values
      > 11 9 7 5 3 X X 3 5
      > 9 11
      > Of course, there are odd divisors higher than 11
      > that may divide 20789 and/or 20791, but let's not bother to check,
      > since it is not germane to this argument

      Au contraire.. there must be and there are odd divisors higher than
      11 that do divide those numbers. The same is true for most all odd
      numbers greater than 11. And one must bother to check them, or at
      least account for them, before a conclusion can be drawn.

      In a nutshell, I think you are focusing on numbers relatively prime
      to n as if all of them were primes, but only a subset of the integers
      relatively prime to n are actually primes. It almost looks that way
      for small n, but in reality, the ratio of primes < n over integers
      less than and relatively prime to n, tends to zero with increasing n.

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