Prime gaps of increasing merit.

[Richard Guy]

Thanks for alerting me to this.  As it still
has question marks round it, it doesn't
properly belong in the following table, but
it's highly likely that it belongs there.
Is it possible that your projected sequence
has a different penultimate term?

  Table 2. Earliest large gaps between consecutive primes.

\begin{tabular}{cccc}
$g$ & $p(g)$ & $(\ln p)^2$ & $g/(\ln p)^2$ \\
456 & 25056082543 & 573.33 & 0.7953 \\
464 & 42652618807 & 599.09 & 0.7745 \\
468 & 127976335139 & 654.09 & 0.7155 \\
474 & 182226896713 & 672.29 & 0.7051 \\
486 & 241160624629 & 686.90 & 0.7075 \\
490 & 297501076289 & 697.95 & 0.7021 \\
500 & 303371455741 & 698.98 & 0.7153 \\
514 & 304599509051 & 699.19 & 0.7351 \\
516 & 416608696337 & 715.85 & 0.7208 \\
532 & 461690510543 & 721.36 & 0.7375 \\
534 & 614487454057 & 736.80 & 0.7247 \\
540 & 738832928467 & 746.84 & 0.7230 \\
582 & 1346294311331 & 779.99 & 0.7462 \\
588 & 1408695494197 & 782.53 & 0.7514 \\
602 & 1968188557063 & 801.35 & 0.7512 \\
652 & 2614941711251 & 817.52 & 0.7975 \\
674 & 7177162612387 & 876.27 & 0.7692 \\
716 & 13829048560417 & 915.53 & 0.7821 \\
766 & 19581334193189 & 936.70 & 0.8178 \\
778 & 42842283926129 & 985.24 & 0.7897 \\
804 & 90874329412297 & 1033.01 & 0.7783 \\
806 & 171231342421327 & 1074.14 & 0.7504 \\
906 & 218209405437449 & 1090.09 & 0.8311 \\
916 & 1189459969826399 & 1204.94 & 0.7602 \\
924 & 1686994940956727 & 1229.32 & 0.7516 \\
1132 & 1693182318747503 & 1229.58 & 0.9206 \\
1184 & 43841547845542243 & 1468.37 & 0.8063 \\
1198 & 55350776431904441 & 1486.29 & 0.8060 \\
1220 & 80873624627234849 & 1515.67 & 0.8049 \\
1442 & 804212830686677669 & 1699.80 & 0.8483
\end{tabular}

It has the second largest entry in the last
column, which noone expects ever to exceed 1.
This is the ``right'' quantity to calculate.
As I've written earlier in the same section
(A8) of UPINT:

It may be worth observing, since the multiplicity
of ln\,s occasionally confuses, that, infinitely often,
the gap $d_n$ exceeds the average gap, $\ln n$, by an
arbitrarily large factor, $M$, say. [Put $n=e^{e^{M^2}}$.]
It's hard to believe that there are infinitely many
gaps $>10^{100}\ln n$.

R.

On Wed, 23 Nov 2005, Ed Pegg Jr wrote:

> Prime gaps of increasing merit = (p_{n+1}-p_n)/log(p_n) 
> might be worthwhile.
> The last term was recently found by Siegfried Herzog & 
> Tomás Oliveira e Silva,
> and is the first increase found in six years.
>
> 2, 1, 1.4427
> 3, 2, 1.82048
> 7, 4, 2.05559
> 113, 14, 2.96147
> 1129, 22, 3.12985
> 1327, 34, 4.72835
> 19609, 52, 5.26116
> 31397, 72, 6.95352
> 155921, 86, 7.19238
> 360653, 96, 7.50254
> 370261, 112, 8.73501
> 1357201, 132, 9.34782
> 4652353, 154, 10.0307
> 2010733, 148, 10.197
> 17051707, 180, 10.8097
> 20831323, 210, 12.4615
> 191912783, 248, 13.003
> 436273009, 282, 14.1753
> 2300942549, 320, 14.8447
> 3842610773, 336, 15.2247
> 4302407359, 354, 15.9586
> 10726904659, 382, 16.5396
> 25056082087, 456, 19.0441
> 304599508537, 514, 19.4386
> 461690510011, 532, 19.8078
> 1346294310749, 582, 20.839
> 1408695493609, 588, 21.0198
> 1968188556461, 602, 21.266
> 2614941710599, 652, 22.8034
> 13829048559701, 716, 23.6633
> 19581334192423, 766, 25.0281
> 218209405436543, 906, 27.4408
> 1693182318746371, 1132, 32.2825
> 804212830686677669, 1442, 34.9757.
>
> http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm#top20merit
>
> --Ed Pegg Jr
>