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Primes and Squares1 - Clarifications2

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  • billoscarson
    On 1/1/04 Andy wrote: One of these days I ll learn to ignore these posts, but until then... It s bad enough working out what the mirror sequence is. It s
    Message 1 of 2 , Jan 2, 2004
      On 1/1/04 Andy wrote:

      One of these days I'll learn to ignore these posts, but until then...

      It's bad enough working out what the mirror sequence is. It's
      impossible
      from the way you've defined it, I had to look at the examples instead.

      So you take all the odd numbers between (2n)^2 and (2n+1)^2, of which
      there are of course 2n. You then write a sequence of pairs:

      {(2n)^2+1, 2n-1},
      {(2n)^2+3, 2n-3},
      {(2n)^2+5, 2n-5},
      ...
      {(2n)^2+2n-3, 3},
      {(2n)^2+2n-1, 1},
      {(2n+1)^2-2n, 1},
      {(2n+1)^2-2n+2,3},
      ...
      {(2n+1)^2-4, 2n-3}
      {(2n+1)^2-2, 2n-1}

      You then claim that, of the 2n terms in this sequence, 2n-2 of them
      consist of 'actual divisors'? Meaning what? What divides what? It's
      not
      clear.

      Andy


      Hi Andy and all:
      Hang in there Andy. You are almost at the end of this
      apparently tortuous path I have imposed upon you, and my apologies
      for that.
      Let me correct a few of your comments. I am studying the odd
      numbers between N^2 and (N+1)^2, not between (2N)^2 and (2N+1)^2.
      You also have more terms in the equations than I do. I don't know
      why, but I hope it will sort itself out from the comments below.
      The first pair should read N^2+1 as the number to be divided, and
      N-1 as the number to divide into it. Don't worry yet that N-1 likely
      won't really divide into N^2+1. Read on to find out why.

      Glad to hear that you got through the definition of "mirror
      sequence" OK. I'll expand and illustrate that concept now for others
      who might not understand what I said, which will lead right into your
      question about what divides what. You can skip to the heading WHAT
      DIVIDES WHAT if you have the mirror sequence under your belt.
      For those of you who are catching up (if anybody), refer to
      message # 14289 for my initial submittal, and message #14307 for the
      clarification of the first part of my arguments, where I left off
      with values for KE and LE for the case N = 12.

      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 of
      all odd integers to N and spans +/- N from that center. Continuing
      with N=12 as the example, the mirror sequence set is centered at
      2*3*5*7*9*11= 20790. The span + or – N from 20790 is 20778 through
      20802. Using odd numbers only, we end up with this series of odd
      numbers in that interval:

      20779 20781 20783 20785 20787 20789 20791 20793 20795
      20797 20799 20801

      Let's call this subset (20779 through 20801) of the complete positive
      integer set the reference set (REF set) just to give it a name.

      Continuing on, we know that 20790 is divisible by all the numbers we
      multiplied to get that product, so we then know that a number 3 away
      from the center of 20790 is divisible by 3 (i.e. 20787 and 20793), a
      number 5 away is divisible by 5, etc. We will want to use only the
      divisors in the LE set, (3,5,7,9,11) since we are still talking about
      N=12. 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 20797
      20799 20801

      LE set values
      11 9 7 5 3 X X 3 5 7
      9 11

      Note that we have used the full LE set of values twice. Note too
      that 11 actually divides 20779 and 20801, 9 actually divides
      20781and 20799, 7 actually divides 20783 and 20797, etc. The two
      X's in the middle indicate there are no odd values in the LE set that
      divide into them. 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 - for N=12, we are limited
      to divisors up to 11.
      MS, the number of actual divisors in the mirror sequence is:
      MS =2*LE, per equation 5 in my initial post.

      WHAT DIVIDES WHAT clarification
      Why have we gone through the above exercise to find divisors of a
      set of numbers from 20779 to 20801 (the REF set), when we are
      analyzing numbers between 145 and 167, and to boot, have chosen to
      not investigate some possible divisors in the REF set? Because we
      are after an upper bound case of divisors in an interval 2N. The
      mirror sequence is the result. The upper bound of divisors in the
      interval will allow us to calculate the lower bound of primes in the
      interval, which is what this argument is all about.

      Now that we have the above sequence of divisors containing 10
      divisors and two spaces (X's), we will use it in the range from 145
      to 167 as the most dense distribution of divisors that could be
      amassed in that range, whether they properly divide numbers in the KE
      set OR NOT. But that's OK. Here's how that looks:

      KE set 145 147 149 151 153 155 157 159 161 163 165
      167 (the set of numbers under study)

      MS set 11 9 7 5 3 X X 3 5 7 9
      11 (the mirror sequence divisors)

      Of course we know that 11 does not divide 145, 9 does not divide
      147, etc (3 happens to divide 153 and 159,and that's OK too, but not
      significant). The point is that we have applied the most dense set
      of divisors to the 2N interval 145 thru 167, yet there are still two
      more numbers to divide than we possibly could have divisors for.
      Therefore we must have at least two primes between these squares.
      This argument holds for all squares.

      Regards, Bill
    • 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 2 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
        mass.

        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
        of
        > 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
        20797
        > 20799 20801
        >
        > LE set values
        > 11 9 7 5 3 X X 3 5
        7
        > 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.


        -Dick
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