Re: Goldbach fine line ... again

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• ... 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
--- 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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