Loading ...
Sorry, an error occurred while loading the content.

Squares & primes1- At least 2 primes between adjacent squares!!?

Expand Messages
  • billoscarson
    Hi All: Here s my take on showing that there are at least two primes between adjacent squares. Let me know what you think. My apologies in advance if the
    Message 1 of 5 , Dec 30, 2003
    • 0 Attachment
      Hi All: Here's my take on showing that there are at least two primes
      between adjacent squares. Let me know what you think. My apologies
      in advance if the tabular examples are scatterred. Has anyone else
      been looking at squares and primes recently?

      ARGUMENTS TO DETERMINE THE MINIMUM NUMBER OF PRIMES BETWEEN TWO
      ADJACENT PERFECT SQUARES

      This article presents a methodology leading to a proof that at
      least two prime numbers exist between any adjacent perfect squares.
      The basic part of this calculation is elementary. The keys to the
      solution are in the observations and assumptions made along the way.

      METHODOLOGY
      First calculate the number of possible primes between any two
      adjacent perfect squares, then calculate the number of possible
      primary divisors for that set. Next use the most dense distribution
      of divisors occurring anywhere in the complete number field, which is
      the "mirror sequence" (ref note 1). Examples are given at the end
      of this article to illustrate the methodology presented here.

      DEFINITIONS
      N^2 = the value of any perfect square, where N is an integer
      from 2 to any N.
      Squares interval = numbers between but not including N^2 and
      (N+1)^2
      Possible primes = all the odd numbers in the squares interval
      Divisor = the denominator in any division operation
      Primary divisors = unique divisor values that are less than or
      equal to N
      KE = the number of possible primes in the interval range if N
      is even
      KD = the number of possible primes in the interval range if N
      is odd
      LE = the number of primary odd divisors less than N, not
      including 1, if N is even
      LD = the number of primary odd divisors less than and including
      N, not including 1, if N is odd
      Mirror sequence = that sequence of primary odd divisors that is
      centered at 2*the product of all
      odd integers to N and spans +/-N from that center.
      MS = the number of actual divisors in the mirror sequence
      J = the minimum number of primes between any two adjacent
      squares

      CALCULATIONS equation number
      KE =
      2*N/2=N
      (1)
      KD = KE-1 =N-1 (ref Note 2)
      (2)
      LE = KE/2 -1 = N/2-1
      (3)
      LD = KD/2 -1 =(N-1)/2-1 (ref Note 2)
      (4)
      For N even:
      MS = 2*LE = 2*(N/2-1) = N-2 (5)
      J = KE – MS = N – (N-2) = 2
      (6) answer
      For N odd:
      MS = 2*LD = 2*[(N-1)/2-1] (7)

      J = KD – MS = N-1 –2*[(N-1)/2-1] = N-1 –(N-1-2) = 2
      (8) answer

      NOTES:
      1. The most dense distribution of divisors is determined by
      examining randomly generated sequences, perturbations of the mirror
      sequence, and actual divisor sets. For all variances, the divisor
      sequence must be internally mathematically consistent, i.e. every
      third odd number must be divisible by 3, every fifth odd number by 5,
      etc. For the purposes of this argument, it is not necessary that a
      divisor in the mirror sequence be a true divisor of the possible
      prime with which it is paired. Using these rules, random sequences
      create spaces when the mirror image set is generated if the original
      set is not in numerical sequence. Further, perturbations of the
      mirror sequence show that any change in the original sequence will
      create one more space. It is possible to create a sequence N-4 long
      when the divisors are not in numerical order. However the mirror
      sequence is N-2 long over the interval N, and thus more dense. These
      findings need to be formally presented before this 2 prime proof is
      complete. Mirror sequences are illustrated in the examples.



      2. KD and LD must be formulated differently than KE and LE to avoid
      round-off inaccuracies
      when calculating J. The reformulation can be made by
      observing that, for odd N's, N^2+2N
      is the last odd number in the squares interval, and is
      divisible by N. Assigning N as divisor
      of N^2+2N then diminishes the available possible primes by 1,
      and also the available primary
      divisors by 1.

      EXAMPLES
      Examples of possible primes (KE or KD) and their mirror sequence
      (MS).

      EVEN N ODD N

      Code: X = no value
      N = 12 N = 13
      KE = 12 MS = 10 KD = 13 MS = 10
      145 11 171 11
      147 9 173 9
      149 7 175 7
      151 5 177 5
      153 3 179 3
      155 X 181 X
      157 X 183 X
      159 3 185 3
      161 5 187 5
      163 7 189 7
      165 9 191 9
      167 11 193 11
      N^2+2N = 195 13*15

      N = 2 N = 3
      KE = 2 MS = 0 KD = 2 MS = 0
      5 X 11 X
      7 X 13 X
      N^2+2N = 15 3*5

      N = 4 N = 5
      KE = 4 MS = 2 KD = 4 MS = 2
      17 3 27 3
      19 X 29 X
      21 X 31 X
      23 3 33 3
      N^2+2N = 35 5*7

      N = 1234E10000, or any even N
      KE = N MS = N-2
      N^2+1 N-1
      N^2+3 N-3
      N^2+5 N-5
      N^2+7 N-7
      * *
      * *
      * *
      X X
      X X
      * *
      * *
      * *
      (N+1)^2-8 N-7
      (N+1)^2-6 N-5
      (N+1)^2-4 N-3
      (N+1)^2-2 N-1

      Examples of calculations
      Even N's
      N 12 90 4566 1234E10000
      KE 12 90 4566 1234E10000
      LE 5 44 2282 617E10000
      MS 10 88 4564 (1234E10000)-2
      J 2 2 2 2

      Odd N's
      N 13 111 9979 123456789123
      KD 12 110 9978 123456789122
      LD 5 54 4988 61728394560
      MS 10 108 9976 123456789120
      J 2 2 2 2

      END
    • Andy Swallow
      Could you re-explain what divisors are? And hence what primary divisors are? It s not clear from what you write, and (3) and (4). I mean a divisor is defined
      Message 2 of 5 , Dec 31, 2003
      • 0 Attachment
        Could you re-explain what divisors are? And hence what primary divisors
        are? It's not clear from what you write, and (3) and (4).

        I mean a divisor is defined as "denominator in any division operation".
        That could be anything at all, couldn't it?

        Presumably you're studying the set J of integers between n^2 and (n+1)^2,
        for n even (there's no need to mess things up by studying even and odd
        separately throughout - just look at the even). By divisors, I assume
        (hope) you mean those integers which divide at least one member of J. Is
        that right? Then the primary divisors are just a certain subset of the
        divisors.

        Please confirm that, or correct me if Im wrong.

        Andy
      • billoscarson
        Hi Andy ; Thanks for taking the time to review my post of 12/30. Here are the answers to your questions. Yes, I am studying the set of integers between N^2
        Message 3 of 5 , Dec 31, 2003
        • 0 Attachment
          Hi Andy ; Thanks for taking the time to review my post of 12/30.

          Here are the answers to your questions.

          Yes, I am studying the set of integers between N^2 and (N+1)^2,
          more specifically the odd integers contained therein, as they are the
          possible prime numbers in that set.
          I agree that we can just look at the evens for now to study the
          technique I am presenting.
          I would call the KE values (for even N) the set of numbers I am
          studying, not J. J is just one number, being equal to the minimum
          number of primes between two squares.
          The fancy footwork associated with the definition of divisors and
          primary divisors is for the purpose of distinguishing between those
          divisors of a number in the KE set that are less than or equal to the
          square root of that number. For example, if a number in the KE set
          were 255, then 3,5 and 17 would be divisors, but only 3 and 5 would
          be primary divisors. I hope this helps. Thanks again for your
          inputs.
        • richard042@yahoo.com
          Hi Bill, I had the same confusion about your intention on divisors and now that I see your reply, another question comes up. If by primary divisors, you mean
          Message 4 of 5 , Dec 31, 2003
          • 0 Attachment
            Hi Bill,

            I had the same confusion about your intention on divisors and now
            that I see your reply, another question comes up. If by primary
            divisors, you mean those that are < sqrt(x), you may have a problem
            with the reasoning because, in the example of x=255, not only are 3 &
            5 primary divisors, but also the trivial 1 & the not so trivial 15
            would be defined as primary divisors. If by primary divisors you
            mean only those divisors that are < sqrt(x) and prime, then the
            definition excludes 15, but what then is the impact on your
            argument?

            -Dick



            --- In primenumbers@yahoogroups.com, "billoscarson"
            <billroscarson@p...> wrote:
            >
            > Hi Andy ; Thanks for taking the time to review my post of 12/30.
            >
            > Here are the answers to your questions.
            >
            > Yes, I am studying the set of integers between N^2 and (N+1)^2,
            > more specifically the odd integers contained therein, as they are
            the
            > possible prime numbers in that set.
            > I agree that we can just look at the evens for now to study the
            > technique I am presenting.
            > I would call the KE values (for even N) the set of numbers I am
            > studying, not J. J is just one number, being equal to the minimum
            > number of primes between two squares.
            > The fancy footwork associated with the definition of divisors
            and
            > primary divisors is for the purpose of distinguishing between those
            > divisors of a number in the KE set that are less than or equal to
            the
            > square root of that number. For example, if a number in the KE set
            > were 255, then 3,5 and 17 would be divisors, but only 3 and 5 would
            > be primary divisors. I hope this helps. Thanks again for your
            > inputs.
          • Andy Swallow
            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
            Message 5 of 5 , Jan 1, 2004
            • 0 Attachment
              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
            Your message has been successfully submitted and would be delivered to recipients shortly.