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Here's some Goldbach separation data

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  • Warren Smith
    GOLDBACH SEPARATION FUNCTION The underlying Goldbach separation data in this table was computed by Tony D. Noe in https://oeis.org/A155765/b155765.txt
    Message 1 of 18 , Sep 18, 2012
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      GOLDBACH SEPARATION FUNCTION
      The underlying "Goldbach separation" data in this table was computed
      by Tony D. Noe in
      https://oeis.org/A155765/b155765.txt
      https://oeis.org/A155764/b155764.txt
      https://oeis.org/A047160
      https://oeis.org/wiki/User:Jason_Kimberley/A047160
      Thanks to Maximilian Hasler for pointing it out.

      For N=1..61, the Nth record-high value of m is printed,
      where m is the least number>=0 such that both a+m and a-m are primes, and
      we keep incrementing a starting from 2 (until a new record-high m is found).
      The final column is f(a) = 0.769422*ln(a)^3. This is my empirical
      upper bound on m, which happens to be valid within the range of the table.
      It can be too high by as much as a factor of 4.715.
      [When m=0 it is infinitely too high, but I ignore that.]

      N a m f(a)
      1, 2, 0, 0.2562370500
      2, 4, 1, 2.049893073
      3, 8, 3, 6.918386629
      4, 22, 9, 22.72372535
      5, 46, 15, 43.18159525
      6, 121, 18, 84.86824603
      7, 128, 21, 87.88911839
      8, 136, 27, 91.22489509
      9, 146, 33, 95.23484236
      10, 238, 39, 126.0861302
      11, 265, 42, 133.6608183
      12, 286, 45, 139.2165668
      13, 341, 48, 152.6127767
      14, 344, 75, 153.3014591
      15, 526, 87, 189.2302088
      16, 904, 93, 242.6605587
      17, 1114, 117, 265.6928937
      18, 1691, 120, 315.9877223
      19, 1736, 135, 319.3490428
      20, 1751, 138, 320.4553082
      21, 1781, 168, 322.6471859
      22, 2476, 183, 367.1566981
      23, 3097, 210, 399.6126911
      24, 3551, 228, 420.3639088
      25, 5131, 300, 479.7382423
      26, 8504, 333, 569.9862949
      27, 10342, 369, 607.7710100
      28, 18526, 393, 730.1603524
      29, 22564, 453, 775.0008925
      30, 24776, 525, 796.8949511
      31, 40072, 621, 915.9875978
      32, 68707, 720, 1063.016969
      33, 125903, 810, 1246.038365
      34, 174913, 846, 1353.652005
      35, 181267, 1086, 1365.690833
      36, 371428, 1281, 1623.109918
      37, 827576, 1305, 1946.678604
      38, 936118, 1515, 1999.977075
      39, 1054141, 1590, 2052.240939
      40, 1159864, 1617, 2094.964675
      41, 1353559, 1722, 2165.244879
      42, 2591107, 1794, 2477.958278
      43, 3759164, 1833, 2670.035444
      44, 5454062, 1851, 2871.819721
      45, 5969581, 2010, 2922.274876
      46, 6363983, 2064, 2958.371376
      47, 6639736, 2085, 2982.466788
      48, 6791669, 2112, 2995.371977
      49, 7183186, 2217, 3027.501431
      50, 9122719, 2352, 3167.106409
      51, 10786495, 2754, 3267.468091
      52, 23848807, 2784, 3771.667809
      53, 27072352, 3339, 3856.745467
      54, 46330868, 3429, 4231.517031
      55, 57989356, 4395, 4395.000000
      56, 183234439, 4488, 5299.392017
      57, 297456022, 5391, 5714.632343
      58, 1010526289, 5430, 6857.998184
      59, 1085549026, 5877, 6929.306766
      60, 1387050394, 6195, 7177.091207
      61, 2117398594, 7107, 7618.514999
    • djbroadhurst
      ... The cube of a log seems reasonable: one log for each member of the pair of primes, on average, and then we throw in an extra log for Tony Noe s extremal
      Message 2 of 18 , Sep 19, 2012
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        --- In primenumbers@yahoogroups.com,
        Warren Smith <warren.wds@...> wrote:

        > The final column is f(a) = 0.769422*ln(a)^3. This is my empirical
        > upper bound on m

        The cube of a log seems reasonable: one log for each member
        of the pair of primes, on average, and then we throw in an
        extra log for Tony Noe's extremal cases, à la Cramér.

        David
      • WarrenS
        ... --that was actually exactly my thinking. However, later, I thought some more. First of all, this picture
        Message 3 of 18 , Sep 19, 2012
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          > > The final column is f(a) = 0.769422*ln(a)^3. This is my empirical
          > > upper bound on m
          >
          > The cube of a log seems reasonable: one log for each member
          > of the pair of primes, on average, and then we throw in an
          > extra log for Tony Noe's extremal cases, à la Cramér.
          >
          > David

          --that was actually exactly my thinking. However, later, I thought some more.
          First of all, this picture
          https://oeis.org/wiki/User:Jason_Kimberley/A047160
          shows what would seem to be a lot of mysterious nonrandom patterns.
          Think they are real?
          So if you believe in them, then Cramer-type randomness-based thinking is out the window, or at least might need to be weakened, e.g. maybe only 1 in every logN guys is
          "random" so you need a 4th power of log not a 3rd power?

          Second, I therefore played around some more trying to find better empirical
          upper bounding functions... I tried f(a) = C * ln(a-1)^B
          and the best parameters I found were B=3.0 and C=0.76942
          which is an empirical upper bound tight within range of table to within
          a factor of 4.70. So, my initial guess seems confirmed in that sense --
          although it remains unhappy in the sense 4.70 is a disappointingly wide range.

          Third, it would be better if the data this all is based on, were extended.
          For that, you'd need a fast prime generating sieve and some other fast
          code for, e.g. symbolically multiplying polynomials of degree 32768 using FFT
          (the polynomials are "generating functions" and the multiplication finds
          "prime coincidences")
          but I would think going out to a=10^14 should be feasible if anybody wanted to
          try hard. The present data only goes out to a few times 10^9.
          Perhaps somebody already did this computation though.
        • WarrenS
          ... --Actually, I think these patterns simply arise from the presence of occasional big prime gaps. Whenever you have one, the two primes at each end of the
          Message 4 of 18 , Sep 19, 2012
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            > this picture
            > https://oeis.org/wiki/User:Jason_Kimberley/A047160
            > shows what would seem to be a lot of mysterious nonrandom patterns.
            > Think they are real?

            --Actually, I think these "patterns" simply arise from the presence of occasional
            big prime gaps. Whenever you have one, the two primes at each end of the gap
            tend to get used a lot in min-discrepancy Goldbach pairs, resulting is a symmetrical
            pair of "45-degree dotted lined segments" in that picture.

            Aside from that effect, it looks patternless. Hence the usual Cramer-type statistical
            model ought to be decently valid, hence my log-cubed bound ought to be
            decent.

            By the way, although I pretty much believe in Cramer-type thinking, there might
            be subtle lower-order nonrandom effects so that when, e.g. Cramer predicts
            the max prime gap below N is log(N)^2, really it ought to be, say,
            C*log(N)^2 for a non-obvious value of C, or log(N)^2 * logloglog(N), or something.
            I don't recommend trusting this kind of thinking to the utmost degree.
            But these kinds of anti-Cramer effects, if any, would be very difficult to spot
            computationally.
          • marku606
            Here s some other Goldbach separation data obtained by looking at the separation with the lowest prime divisors. 2N = p1 + p2. We look at the factors of p2 -
            Message 5 of 18 , Sep 21, 2012
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              Here's some other Goldbach separation data obtained by looking at the separation with the lowest prime divisors.


              2N = p1 + p2. We look at the factors of p2 - p1 such that the largest prime which divides p2 - p1 is minimal.

              For instance,

              4 = 2 + 2 ; 2 - 2 = 0
              8 = 3 + 5 ; 5 - 3 = 2
              16 = 5 + 11 ; 11 - 5 = 2*3
              48 = 19 + 29 ; 29 - 19 = 2*5


              Here are the results for 2n up to 100,000: [2n, minimal largest prime that divides p2-p1]

              [4, 0] [8, 2] [16, 3] [48, 5] [180, 7] [420, 11] [1680, 13] [21840, 17] [27300, 19] [78540, 23]


              The minimal largest prime seems to be approximated by log(2n)^1.3 but I have not looked very far.


              Mark
            • djbroadhurst
              ... Definition: For prime p, let n(p) be the smallest integer n 1 for which there is no pair of primes [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is
              Message 6 of 18 , Sep 22, 2012
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                --- In primenumbers@yahoogroups.com,
                "marku606" <mark.underwood@...> wrote:

                > 2N = p1 + p2. We look at the factors of p2 - p1 such that
                > the largest prime which divides p2 - p1 is minimal.

                > Here are the results for 2n up to 100,000:
                > [2n, minimal largest prime that divides p2-p1]

                > [4, 0] [8, 2] [16, 3] [48, 5] [180, 7] [420, 11]
                > [1680, 13] [21840, 17] [27300, 19] [78540, 23]

                Definition: For prime p, let n(p) be the smallest
                integer n > 1 for which there is no pair of primes
                [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.

                Examples: [p, n(p), factor(n(p))]

                [2, 8, Mat([2, 3])]
                [3, 24, [2, 3; 3, 1]]
                [5, 90, [2, 1; 3, 2; 5, 1]]
                [7, 210, [2, 1; 3, 1; 5, 1; 7, 1]]
                [11, 840, [2, 3; 3, 1; 5, 1; 7, 1]]
                [13, 10920, [2, 3; 3, 1; 5, 1; 7, 1; 13, 1]]
                [17, 13650, [2, 1; 3, 1; 5, 2; 7, 1; 13, 1]]
                [19, 39270, [2, 1; 3, 1; 5, 1; 7, 1; 11, 1; 17, 1]]
                [23, 1492260, [2, 2; 3, 1; 5, 1; 7, 1; 11, 1; 17, 1; 19, 1]]

                Puzzle: Find n(29).

                Conjecture: n(p) is p-smooth for all prime p.

                David
              • djbroadhurst
                ... Puzzle 31: Prove that n(31)
                Message 7 of 18 , Sep 23, 2012
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                  --- In primenumbers@yahoogroups.com,
                  "djbroadhurst" <d.broadhurst@...> wrote:

                  > Definition: For prime p, let n(p) be the smallest
                  > integer n > 1 for which there is no pair of primes
                  > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.
                  >
                  > Puzzle: Find n(29).

                  Puzzle 31: Prove that n(31) < 3*10^8.

                  David
                • djbroadhurst
                  ... Puzzle 37: Prove that n(37)
                  Message 8 of 18 , Sep 23, 2012
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                    --- In primenumbers@yahoogroups.com,
                    "djbroadhurst" <d.broadhurst@...> wrote:

                    > Definition: For prime p, let n(p) be the smallest
                    > integer n > 1 for which there is no pair of primes
                    > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.
                    >
                    > Puzzle: Find n(29).
                    >
                    > Puzzle 31: Prove that n(31) < 3*10^8.

                    Puzzle 37: Prove that n(37) < 2*10^10.

                    Comment: This may suggest a "hard" new
                    OEIS sequence, when the answers are in.

                    David
                  • Maximilian Hasler
                    ... It seems that n(31) is the smallest member of a set including n = 281291010 = 29#/23 (?) Maximilian
                    Message 9 of 18 , Sep 23, 2012
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                      On Sun, Sep 23, 2012 at 11:31 AM, djbroadhurst <d.broadhurst@...> wrote:
                      >
                      > > Definition: For prime p, let n(p) be the smallest
                      > > integer n > 1 for which there is no pair of primes
                      > > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.
                      >
                      > Puzzle 31: Prove that n(31) < 3*10^8.


                      It seems that n(31) is the smallest member of a set
                      including n = 281291010 = 29#/23 (?)

                      Maximilian
                    • Maximilian Hasler
                      ... Hint: don t search beyond 11741730 = 23# / 19. Maximilian
                      Message 10 of 18 , Sep 23, 2012
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                        > On Sun, Sep 23, 2012 at 11:31 AM, djbroadhurst <d.broadhurst@...> wrote:
                        >>
                        >> > Definition: For prime p, let n(p) be the smallest
                        >> > integer n > 1 for which there is no pair of primes
                        >> > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.
                        >>
                        >> Puzzle: Find n(29).

                        Hint: don't search beyond 11741730 = 23# / 19.

                        Maximilian
                      • djbroadhurst
                        ... Indeed. Moreover, I have shown, by exhaustion, that ... for which my personal mnemonic is the Kupferbach conjecture . The Kupferbach and Goldbach
                        Message 11 of 18 , Sep 23, 2012
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                          --- In primenumbers@yahoogroups.com,
                          Maximilian Hasler <maximilian.hasler@...> wrote:

                          > > Definition: For prime p, let n(p) be the smallest
                          > > integer n > 1 for which there is no pair of primes
                          > > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.
                          >
                          > > Puzzle 31: Prove that n(31) < 3*10^8.
                          >
                          > It seems that n(31) is the smallest member of a set
                          > including n = 281291010 = 29#/23 (?)

                          Indeed. Moreover, I have shown, by exhaustion, that
                          281291010 is the smallest value of n(31) consistent with:

                          > Conjecture: n(p) is p-smooth for all prime p.

                          for which my personal mnemonic is the "Kupferbach conjecture".

                          The "Kupferbach" and Goldbach conjectures seem to me to be
                          logically independent: a resolution of either of them does
                          not seem to yield an immediate resolution of the other.

                          While "Kupferbach" is supported by far less evidence,
                          yet it might be easier to prove than Goldbach?

                          David
                        • djbroadhurst
                          ... Congrats for the correct answer. I had already proven, unconditionally, that n(29) = 11741730, so that s one more for OEIS, if Maximilian would like to
                          Message 12 of 18 , Sep 23, 2012
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                            --- In primenumbers@yahoogroups.com,
                            Maximilian Hasler <maximilian.hasler@...> wrote:

                            > > Definition: For prime p, let n(p) be the smallest
                            > > integer n > 1 for which there is no pair of primes
                            > > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.
                            >
                            > > Puzzle: Find n(29).
                            >
                            > Hint: don't search beyond 11741730 = 23# / 19.

                            Congrats for the correct answer. I had already proven,
                            unconditionally, that n(29) = 11741730, so that's one
                            more for OEIS, if Maximilian would like to draft an entry.

                            With thanks to Mark for the idea,

                            David
                          • Mark
                            My thanks to David for taking an idea to the next level, brilliant. I ran some laggardly code last night and only this morning did I arrive at the 11741730
                            Message 13 of 18 , Sep 24, 2012
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                              My thanks to David for taking an idea to the next level, brilliant. I ran some laggardly code last night and only this morning did I arrive at the 11741730 figure.

                              I remember (was it last year) doing some investigations on the counts of p smooth numbers, and it was David who again took it to the next level by arriving at formulae to compute exact counts, if I recall. The King of Smooth in my mind. :)

                              Mark


                              --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
                              >
                              >
                              >
                              > --- In primenumbers@yahoogroups.com,
                              > Maximilian Hasler <maximilian.hasler@> wrote:
                              >
                              > > > Definition: For prime p, let n(p) be the smallest
                              > > > integer n > 1 for which there is no pair of primes
                              > > > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.
                              > >
                              > > > Puzzle: Find n(29).
                              > >
                              > > Hint: don't search beyond 11741730 = 23# / 19.
                              >
                              > Congrats for the correct answer. I had already proven,
                              > unconditionally, that n(29) = 11741730, so that's one
                              > more for OEIS, if Maximilian would like to draft an entry.
                              >
                              > With thanks to Mark for the idea,
                              >
                              > David
                              >
                            • djbroadhurst
                              ... An OEIS sequence 8, 24, 90, 210, 840, 10920, 13650, 39270, 1492260, 11741730 has been proposed by Maximilian and awaits approval. Thereafter we have only
                              Message 14 of 18 , Sep 24, 2012
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                                --- In primenumbers@yahoogroups.com,
                                "Mark" <mark.underwood@...> wrote:

                                > I ran some laggardly code last night and only this
                                > morning did I arrive at the 11741730 figure.

                                An OEIS sequence

                                8, 24, 90, 210, 840, 10920, 13650, 39270, 1492260, 11741730

                                has been proposed by Maximilian and awaits approval.

                                Thereafter we have only the upper bounds

                                n(31) <= 281291010 = 19#*29
                                n(37) <= 10919808900 = 5#*17#*23*31

                                where

                                > Definition: For prime p, let n(p) be the smallest
                                > integer n > 1 for which there is no pair of primes
                                > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.

                                The upper limit for p = 31 is tight on the fairly
                                reasonable assumption that n(31) is 31-smooth.

                                The upper limit for p = 37 is tight on the rather
                                stronger assumption that n(37) is both 37-smooth
                                and also divisible by 17#.

                                It is instructive to study the Goldbach partitions,
                                n +/- m, in the case n = 281291010, where there is no
                                partition for which m is 31-smooth. The near misses
                                are these 37-smooth values of m:

                                [19573, [23, 2; 37, 1]]
                                [450179, [23, 3; 37, 1]]
                                [817811, [23, 1; 31, 2; 37, 1]]
                                [1165019, [23, 1; 37, 3]]
                                [1570243, [31, 1; 37, 3]]
                                [1874161, Mat([37, 4])]
                                [13955549, [23, 3; 31, 1; 37, 1]]
                                [16656623, [23, 3; 37, 2]]
                                [36115589, [23, 1; 31, 1; 37, 3]]

                                Since we require gcd(m,n) = 1 for the primality of n +/- m,
                                the only primes less than 37 available to m are 23 and 31,
                                neither of which were used for this amusing Goldbach pair:

                                19#*29+37^4 is prime
                                19#*29-37^4 is prime

                                It took Pari-GP about 2020 GHz-seconds to prove that there
                                is no 37-smooth value of m in the case n = 10919808900.
                                The near misses are these 41-smooth values of m:

                                [52307677, [29, 2; 37, 1; 41, 2]]
                                [66737381, [29, 1; 37, 2; 41, 2]]
                                [76840601, [37, 4; 41, 1]]
                                [197696957, [19, 4; 37, 1; 41, 1]]
                                [1680914269, [29, 3; 41, 3]]
                                [4493598401, [19, 4; 29, 2; 41, 1]]

                                which may also use the primes 19, 29 and 37, omitted
                                from n. Note that 19 and 37 do not appear here:

                                5#*17#*23*31+(29*41)^3 is prime
                                5#*17#*23*31-(29*41)^3 is prime

                                David
                              • Robert Gerbicz
                                Thereafter we have only the upper bounds n(31)
                                Message 15 of 18 , Sep 25, 2012
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                                  "Thereafter we have only the upper bounds

                                  n(31) <= 281291010 = 19#*29
                                  n(37) <= 10919808900 = 5#*17#*23*31"

                                  Using exhaustive search there is no better solution, so n(31)=**281291010
                                  and n(37)=10919808900. (and also confirmed smaller n() values).

                                  First sieve out n, where 2n=p1+p2 and p1<=p2<=p1+bound and p2-p1 is
                                  p-smooth (you can choose say bound=20000). For the survivors used
                                  backtracking to generate possible values for d=p2-p1 (where d is p-smooth)
                                  and done primality tests for p1 and p2. It took 3 miutes to find n(31) and
                                  12 hours for n(37).


                                  [Non-text portions of this message have been removed]
                                • WarrenS
                                  ... --Equivalent easier-to-read definition: For prime q, let n(q) be the least integer n 1 for which every pair of primes [p1, p2] with 2*n=p1+p2 has p2-p1
                                  Message 16 of 18 , Sep 25, 2012
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                                    > Definition: For prime p, let n(p) be the smallest
                                    > integer n > 1 for which there is no pair of primes
                                    > [p1, p2] such that 2*n = p1 + p2 and p2 - p1 is p-smooth.

                                    --Equivalent easier-to-read definition:
                                    For prime q, let n(q) be the least
                                    integer n>1 for which every pair of primes
                                    [p1, p2] with 2*n=p1+p2 has
                                    p2-p1 having a prime factor exceeding q.

                                    > Conjecture: n(q) is q-smooth for all prime q.
                                    > for which my personal mnemonic is the "Kupferbach conjecture".

                                    > The "Kupferbach" and Goldbach conjectures seem to me to be
                                    > logically independent: a resolution of either of them does
                                    > not seem to yield an immediate resolution of the other.

                                    --Note: if Goldbach conjecture is false, then the counterexample n,
                                    such that 2n is not the sum of two primes, will be the value of n(q)
                                    for all large-enough q. Thus Kupferbach will be asymptotically true.
                                    So no, they are not logically independent, at least as stated.
                                    (You may have intended a somewhat different statement.)

                                    > While "Kupferbach" is supported by far less evidence,
                                    > yet it might be easier to prove than Goldbach?
                                    > David

                                    --it might be a new approach to Goldbach which might say something interesting.
                                    I am not exactly an expert on the Goldbach conjecture but this Kupferbach thing
                                    is rather a different angle than anything Goldbach I saw before.

                                    I had wanted that Goldbach separation data because I happen to actually have a use for the Goldbach conjecture (!) and in that use, this "separation" way of quantifying it, is what matters.
                                  • djbroadhurst
                                    ... Kupferbach conjectures that n(p) is p-smooth for /all/ prime p. A failure of Goldbach at N would tell us /nothing/ about Kupferbach in the cases with
                                    Message 17 of 18 , Sep 25, 2012
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                                      --- In primenumbers@yahoogroups.com,
                                      "WarrenS" <warren.wds@...> wrote:

                                      > --Note: if Goldbach conjecture is false, then the counterexample n,
                                      > such that 2n is not the sum of two primes, will be the value of n(q)
                                      > for all large-enough q. Thus Kupferbach will be asymptotically
                                      > true.

                                      "Kupferbach" conjectures that n(p) is p-smooth for /all/ prime p.

                                      A failure of Goldbach at N would tell us /nothing/ about "Kupferbach"
                                      in the cases with n(p) < N.

                                      Hence I maintain my stated opinion:

                                      > The "Kupferbach" and Goldbach conjectures seem to me to be
                                      > logically independent: a resolution of either of them does
                                      > not seem to yield an immediate resolution of the other.

                                      David
                                    • djbroadhurst
                                      ... Thanks, Robert. Now Maximilian can update http://oeis.org/A217016 David
                                      Message 18 of 18 , Sep 25, 2012
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                                        --- In primenumbers@yahoogroups.com, Robert Gerbicz <robert.gerbicz@...> wrote:

                                        > Using exhaustive search there is no better solution,
                                        > so n(31)=281291010 and n(37)=10919808900.

                                        Thanks, Robert. Now Maximilian can update http://oeis.org/A217016

                                        David
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