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Cumulative sum of Goldbach Partitions

  1. Sep 23, 2011 #1
    I noted the following concerning the cumulative sum of Goldbach partitions


    C[2N] = sum[ G(2N) ;from 6 to 2N] is greater than pi[2N]*(pi[2N] -1)/2

    where
    2N is an even number 2N=6,,,,,
    C[2N] is the cumulative sum of the goldbach partitions of the even numbers 6,...2N
    G(2N) is the number of Goldbach partitions for the even number 2N and
    pi[2N] is the number of primes less than 2N.

    I checked the above on my computer for values of 2N upto 100,000

    What would a proof of this be? I got as far as formulating every possible combinations
    of primes (p,q) where p and q are both less than 2N, This lead me to a rough guess of
    C[2N] = pi^2[2N] /2 but why is it greater?.

    Paul
     
  2. jcsd
  3. Sep 24, 2011 #2
    I have been able to prove a weak version of the above.

    Let the cumulative sum C[2N] = Sum([G[2N]; 2N= 6 to 2N)
    where G[2N] is the number of goldbach partitions for the even number 2N.

    Consider every combination of odd primes less than N,
    note that the sum of these combination pairs is less than 2N
    and note that this set of combination pairs forms a subset
    of all combination pairs of odd primes that form the goldbach
    partitions constituting the cumulative sum thus
    we can state

    pi[N]*pi[N] < C[2N]

    where pi[N] is the number of primes less than N

    though I still am unable to prove

    pi[2N]*(pi[2N]-1)/2 < C[2N]

    the latter is closer to the function C[2N]

    I found another function which appears to be
    even more closer to C[2N] for large 2N,
    though it does not hold for some small 2N


    pi[2N]*pi[N] < C[2N]

    Any suggestions?
     
  4. Sep 25, 2011 #3
    To get familiar with this topic, I started to establish the function
    Goldbach2 to calculate the number of partitons of 2n; the result
    r2(2n) is the number of prime pairs (p,q) with p not greater q;
    up to 40 the list looks like this:

    r(2n)|2n|Sum(r(2n))
    -------------------
    1|4|1
    1|6|2
    1|8|3
    2|10|5
    1|12|6
    2|14|8
    2|16|10
    2|18|12
    2|20|14

    Perhaps, you did not use r2(n), but R(2n), which is simply 2*r(2n) if
    n is not prime, and 2*r(2n) - 1 else.

    Regards
     
  5. Sep 25, 2011 #4
    To better understand the terminology I have included a word document with a graph with values 2N= 6 to 72,000. It appears the term R[2N] you refered to and G[2N] are interchangeable.

    Regards

    Paul
     

    Attached Files:

  6. Sep 26, 2011 #5
    Code (Text):

    Below is a table with the following entries:


    R2(2*n) is definied as in my previous post as the
            number of binary Goldbach partitions
            (eg R2(10) = 3; the partiitions are {(3+7),(5+5),(7+3)}

    C(2*n) := Sum(R2(2*n)), the sum is taken from 4 to 2*n
    Pi(k) are the number of primes not exceeding k
    H1 := Pi(n) * Pi(n)
    H2 := Pi(n) * Pi(2*n)
    H3 := Pi(2*n)*(Pi(2*n) - 1) / 2

        2*n    C(2*n)    Pi(n)  Pi(2*n)      H1       H2       H3
    -------------------------------------------------------------
          4        1        1        2        1        2        1
          6        2        2        3        4        6        3
          8        4        2        4        4        8        6
         10        7        3        4        9       12        6
         20       24        4        8       16       32       28
         30       50        6       10       36       60       45
         40       78        8       12       64       96       66
         50      117        9       15       81      135      105
         60      158       10       17      100      170      136
         70      199       11       19      121      209      171
         80      252       12       22      144      264      231
         90      312       14       24      196      336      276
        100      361       15       25      225      375      300
        200     1135       25       46      625     1150     1035
        300     2233       35       62     1225     2170     1891
        400     3540       46       78     2116     3588     3003
        500     5139       53       95     2809     5035     4465
        600     6946       62      109     3844     6758     5886
        700     9012       70      125     4900     8750     7750
        800    11274       78      139     6084    10842     9591
        900    13759       87      154     7569    13398    11781
       1000    16349       95      168     9025    15960    14028
       2000    52934      168      303    28224    50904    45753
       3000   105975      239      430    57121   102770    92235
       4000   173401      303      550    91809   166650   150975
       5000   254879      367      669   134689   245523   223446
       6000   349452      430      783   184900   336690   306153
       7000   456847      489      900   239121   440100   404550
       8000   576324      550     1007   302500   553850   506521
       9000   707138      610     1117   372100   681370   623286
      10000   850833      669     1229   447561   822201   754606

    H1, H2 and H3 serve as lower estimates for C2(2*n)
    and H2 seems to serve best

    Regards
     
     
  7. Sep 26, 2011 #6
    Hi

    I think I have found an upper bound for C[2N] [though it does not hold for some small 2N]. Again an intuitive guess.

    C[2N] < 4*Pi[2N]*Pi[N] - 2*Pi[N]*Pi[N] - P[2N]*Pi[2N]

    so we have

    Pi[N]*Pi[2N] < C[2N] < 4*Pi[2N]*Pi[N] - 2*Pi[N]*Pi[N] - P[2N]*Pi[2N]

    I checked this for odd primes upto 2N= 100,000

    Regards
     
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