Prime pair points slope approaches 1
Solution 1:
It is really just that prime gaps remain quite small. If $q$ is the next prime larger than $p,$ then for all known primes, $$ q < p + 3 \log^2 p. $$ This is currently unprovable but consistent with Cramer's conjecture. All that can be proved is something like $q < p + p^{0.55}$ for sufficiently large $p,$ i don't think there is even a known bound for beginning of that behavior. So proof is far behind observation.
Here is a nice table of prime gaps. Since the merit is the gap over $\log p,$ all one needs to do to get the gap divided by $\log^2 p$ is divide by another $\log p.$ They tell you the number of digits in each P, and then $\log P$ is about 2.3 times $\log_{10} P.$ It would be nice HINT HINT to see this big table with each gap divided by $\log^2 p,$ as Guy does below.
Richard K. Guy, in Unsolved Problems in Number Theory, gives the gaps divided by $\log^2 p.$ In the second edition (1994), page 21, in the third edition (2004), page 36.
Stolen from
http://users.cybercity.dk/~dsl522332/math/primegaps/maximal.htm
the size of the gap is g
next are the number of decimal digits in p
for 4 * 10^18 > p >= 11, g < log^2 p = (log p)^2.
Oh, logarithms base e == 2.718281828459
==================================
g digits of p p log p g/log p g/log^2 p
1 1 1 2 0.693147 1.4427 2.08137
2 2 1 3 1.09861 1.82048 1.65707
3 4 1 7 1.94591 2.05559 1.05637
4 6 2 23 3.13549 1.91357 0.610294
5 8 2 89 4.48864 1.78228 0.397065
6 14 3 113 4.72739 2.96147 0.626449
7 18 3 523 6.25958 2.87559 0.45939
8 20 3 887 6.78784 2.94644 0.434076
9 22 4 1129 7.02909 3.12985 0.445271
10 34 4 1327 7.19068 4.72835 0.657566
11 36 4 9551 9.1644 3.92824 0.428642
12 44 5 15683 9.66033 4.55471 0.471486
13 52 5 19609 9.88374 5.26116 0.532305
14 72 5 31397 10.3545 6.95352 0.671548
15 86 6 155921 11.9571 7.19238 0.601515
16 96 6 360653 12.7957 7.50254 0.586334
17 112 6 370261 12.822 8.73501 0.681254
18 114 6 492113 13.1065 8.698 0.663642
19 118 7 1349533 14.1153 8.35974 0.592248
20 132 7 1357201 14.1209 9.34782 0.661983
21 148 7 2010733 14.514 10.197 0.702566
22 154 7 4652353 15.3529 10.0307 0.653342
23 180 8 17051707 16.6518 10.8097 0.649161
24 210 8 20831323 16.852 12.4615 0.739466
25 220 8 47326693 17.6726 12.4487 0.704405
26 222 9 122164747 18.6209 11.9221 0.640254
27 234 9 189695659 19.0609 12.2764 0.644062
28 248 9 191912783 19.0726 13.003 0.681764
29 250 9 387096133 19.7742 12.6427 0.639356
30 282 9 436273009 19.8938 14.1753 0.712549
31 288 10 1294268491 20.9812 13.7266 0.654231
32 292 10 1453168141 21.097 13.8408 0.656056
33 320 10 2300942549 21.5566 14.8447 0.688637
34 336 10 3842610773 22.0694 15.2247 0.689855
35 354 10 4302407359 22.1824 15.9586 0.719423
36 382 11 10726904659 23.096 16.5396 0.716125
37 384 11 20678048297 23.7523 16.1668 0.680642
38 394 11 22367084959 23.8309 16.5332 0.693772
39 456 11 25056082087 23.9444 19.0441 0.795349
40 464 11 42652618343 24.4764 18.9571 0.774506
41 468 12 127976334671 25.5751 18.299 0.715502
42 474 12 182226896239 25.9285 18.281 0.705055
43 486 12 241160624143 26.2087 18.5434 0.707529
44 490 12 297501075799 26.4187 18.5475 0.702059
45 500 12 303371455241 26.4382 18.912 0.715328
46 514 12 304599508537 26.4423 19.4386 0.735133
47 516 12 416608695821 26.7554 19.2858 0.720819
48 532 12 461690510011 26.8582 19.8078 0.737495
49 534 12 614487453523 27.1441 19.6728 0.724756
50 540 12 738832927927 27.3283 19.7597 0.723048
51 582 13 1346294310749 27.9284 20.839 0.746159
52 588 13 1408695493609 27.9737 21.0198 0.751412
53 602 13 1968188556461 28.3081 21.266 0.751232
54 652 13 2614941710599 28.5923 22.8034 0.797536
55 674 13 7177162611713 29.6019 22.7688 0.769166
56 716 14 13829048559701 30.2578 23.6633 0.782057
57 766 14 19581334192423 30.6056 25.0281 0.817762
58 778 14 42842283925351 31.3885 24.7861 0.789655
59 804 14 90874329411493 32.1405 25.0152 0.778307
60 806 15 171231342420521 32.774 24.5926 0.750369
61 906 15 218209405436543 33.0165 27.4408 0.831126
62 916 16 1189459969825483 34.7123 26.3884 0.760203
63 924 16 1686994940955803 35.0617 26.3535 0.751632
64 1132 16 1693182318746371 35.0654 32.2825 0.920639
65 1184 17 43841547845541059 38.3194 30.8982 0.806335
66 1198 17 55350776431903243 38.5525 31.0745 0.806032
67 1220 17 80873624627234849 38.9317 31.337 0.804922
68 1224 18 203986478517455989 39.8568 30.7099 0.770506
69 1248 18 218034721194214273 39.9234 31.2598 0.782995
70 1272 18 305405826521087869 40.2604 31.5943 0.784749
71 1328 18 352521223451364323 40.4039 32.8681 0.813489
72 1356 18 401429925999153707 40.5338 33.4536 0.825325
73 1370 18 418032645936712127 40.5743 33.7652 0.832181
74 1442 18 804212830686677669 41.2286 34.9757 0.848335
75 1476 19 1425172824437699411 41.8008 35.3103 0.844728
g digits of p p log p g/log p g/log^2 p
==================================