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Re: Goldbach fine line ... again

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  • Mark Underwood
    ... Great data! Thank you Jens. And here I was fiddling with formula based on my own numbers merely up to 10^8. Well - and I am somewhat surprised - so far
    Message 1 of 3 , Nov 23, 2008
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      --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
      >
      > Mark Underwood wrote:
      > > Given that every even number n greater than two can be written as
      > > the sum of two primes, is at least one of the smallest primes
      > > always less than ( (log(n+2))^e)/2 ?
      >
      > For even n > 2, let p(n) be the smallest prime p such n-p is prime.
      > Let f(n) = ((log(n+2))^e)/2.
      > Mark asks whether p(n) < f(n) for all n. It holds for n < 1.25*10^18.
      >
      > Tomás Oliveira e Silva has a Goldbach conjecture verification project
      > at http://www.ieeta.pt/~tos/goldbach.html. It links a table which
      > currently includes all n < 1.25*10^18 (double tested to 10^17) such
      > that p(n) is larger than for any smaller n.
      > If p(n) >= f(n) ever occurs then the smallest n will be of the
      > mentioned type.
      >
      > n and p(n) are from Silva's table.
      > The f(n) column is rounded to the nearest integer.
      >
      > . n p(n) f(n) f(n)/p(n)
      > . 4 2 2 1.22
      > . 6 3 4 1.22
      > . 12 5 7 1.40
      > . 30 7 15 2.10
      > . 98 19 32 1.67
      > . 220 23 49 2.13
      > . 308 31 58 1.86
      > . 556 47 75 1.60
      > . 992 73 95 1.31
      > . 2642 103 137 1.33
      > . 5372 139 173 1.24
      > . 7426 173 191 1.10
      > . 43532 211 313 1.48
      > . 54244 233 330 1.42
      > . 63274 293 343 1.17
      > . 113672 313 395 1.26
      > . 128168 331 406 1.23
      > . 194428 359 447 1.24
      > . 194470 383 447 1.17
      > . 413572 389 526 1.35
      > . 503222 523 548 1.05
      > . 1077422 601 639 1.06
      > . 3526958 727 798 1.10
      > . 3807404 751 809 1.08
      > . 10759922 829 969 1.17
      > . 24106882 929 1105 1.19
      > . 27789878 997 1131 1.13
      > . 37998938 1039 1188 1.14
      > . 60119912 1093 1275 1.17
      > . 113632822 1163 1401 1.21
      > . 187852862 1321 1507 1.14
      > . 335070838 1427 1635 1.15
      > . 419911924 1583 1686 1.07
      > . 721013438 1789 1814 1.01
      > . 1847133842 1861 2051 1.10
      > . 7473202036 1877 2437 1.30
      > . 11001080372 1879 2551 1.36
      > . 12703943222 2029 2595 1.28
      > . 21248558888 2089 2753 1.32
      > . 35884080836 2803 2922 1.04
      > . 105963812462 3061 3289 1.07
      > . 244885595672 3163 3592 1.14
      > . 599533546358 3457 3936 1.14
      > . 3132059294006 3463 4623 1.33
      > . 3620821173302 3529 4686 1.33
      > . 4438327672994 3613 4776 1.32
      > . 5320503815888 3769 4858 1.29
      > . 8342945544436 3917 5063 1.29
      > . 10591605900482 4003 5174 1.29
      > . 12982270197518 4027 5270 1.31
      > . 15197900994218 4057 5345 1.32
      > . 28998050650046 4327 5660 1.31
      > . 46878442766282 4519 5902 1.31
      > . 76903574497118 4909 6158 1.25
      > . 184162477860248 5077 6626 1.31
      > . 217361316706568 5209 6717 1.29
      > . 389965026819938 5569 7045 1.27
      > . 1047610575836828 6469 7623 1.18
      > . 6253262345930828 6961 8741 1.26
      > . 24925556008175266 7559 9674 1.28
      > . 31284177910528922 7753 9833 1.27
      > . 121005022304007026 8443 10815 1.28
      > . 255329126688555994 8501 11382 1.34
      > . 258549426916149682 8933 11392 1.28
      > . 555274351556750822 8941 11992 1.34
      > . 887123803077837868 9161 12369 1.35
      > . 906030579562279642 9341 12386 1.33
      >
      > The question is whether the last column will ever go below 1.
      > It has already been below 1.1 many times, but not for a while.
      >


      Great data! Thank you Jens. And here I was fiddling with formula based on my own
      numbers merely up to 10^8. Well - and I am somewhat surprised - so far so good. The
      padding between the data and the formula bounds seems to be getting consistently
      thicker past around 10^17. But who knows, that could change on a dime, knowing primes.

      Mark
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