As it is well known that prime number is $2,3,5\cdots \cdots$, thus all these prime number are denoted by$p_{1},p_{2},\cdots \cdots ,p_{n}\cdots \cdots$. The prime maximal gap $\max_{p_{n+1}\leqslant N}(p_{n+1}-p_{n})$ means the maximum value of $ (p_{2}-p_{1},p_{3}-p_{2},\cdots \cdots ,p_{n+1}-p_{n})$. In 1937, Cramér gave a conjecture about the prime maximal gaps that $$\lim_{n\rightarrow \infty }sup\frac{p_{n+1}-p_{n}}{(logp_{n})^{2}}=1$$which is still an unproven conjecture.
**I found a conjecture about the prime maximal gaps that $$\max_{p_{n+1}\leqslant N }(p_{n+1}-p_{n})\approx logN(logN-2loglogN)+2$$ when $N\geqslant 7$.** My conjecture gives an approximate value of the prime maximal gap ,which is close to the actual value.
**question: Has anyone a clue how to prove or disprove the above conjecture?**
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\begin{matrix}
A& B & C & D & E & F & G\\\
1&2&1&——& ——& ——& ——\\\
2 & 3 & 2 &—— & —— & —— & ——\\\
3 &7 &4 & 3 & 0.75 & 4 & 1.00\\\
4 & 23 & 6 & 5 & 0.83 & 10 & 1.67\\\
5& 89& 8& 9& 1.13& 20& 2.50\\\
6& 113& 14& 10& 0.71& 22& 1.57\\\
7& 523& 18& 18& 1.00& 39& 2.17\\\
8& 887& 20& 22& 1.10& 46& 2.30\\\
9& 1129& 22& 24& 1.09& 49& 2.23\\\
10& 1327& 34& 25& 0.74& 52& 1.53\\\
11& 9551& 36& 45& 1.25& 84& 2.33\\\
12& 15683& 44& 51& 1.16& 93& 2.11\\\
13& 19609& 52& 54& 1.04& 98& 1.88\\\
14& 31397& 72& 61& 0.85& 107& 1.49\\\
15& 155921& 86& 86& 1.00& 143& 1.66\\\
16& 360653& 96& 100& 1.04& 164& 1.71\\\
17& 370261& 112& 101& 0.90& 164& 1.46\\\
18& 492113& 114& 106& 0.93& 172& 1.51\\\
19& 1349533& 118& 127& 1.08& 199& 1.69\\\
20& 1357201& 132& 127& 0.96& 199& 1.51\\\
21& 2010733& 148& 135& 0.91& 211& 1.43\\\
22& 4652353& 154& 154& 1.00& 236& 1.53\\\
23& 17051707& 180& 186& 1.03& 277& 1.54\\\
24& 20831323& 210& 191& 0.91& 284& 1.35\\\
25& 47326693& 220& 213& 0.97& 312& 1.42\\\
26& 122164747& 222& 240& 1.08& 347& 1.56\\\
27& 189695659& 234& 253& 1.08& 363& 1.55\\\
28& 191912783& 248& 253& 1.02& 364& 1.47\\\
29& 387096133& 250& 275& 1.10& 391& 1.56\\\
30& 436273009& 282& 279& 0.99& 396& 1.40\\\
31& 1294268491 &288& 314& 1.09& 440& 1.53\\\
32& 1453168141& 292& 318& 1.09& 445& 1.52\\\
33& 2300942549& 320& 334& 1.04& 465& 1.45\\\
34& 3842610773 &336& 352& 1.05& 487& 1.45\\\
35& 4302407359& 354& 357& 1.01& 492& 1.39\\\
36& 10726904659& 382& 390& 1.02& 533& 1.40\\\
37& 20678048297& 384& 416& 1.08& 564& 1.47\\\
38& 22367084959& 394& 419& 1.06& 568& 1.44\\\
39& 25056082087& 456& 423& 0.93& 573& 1.26\\\
40& 42652618343& 464& 445& 0.96& 599& 1.29\\\
41& 127976334671& 468& 490& 1.05& 654& 1.40\\\
42& 182226896239& 474& 505& 1.07& 672& 1.42\\\
43& 241160624143& 486& 518& 1.07& 687& 1.41\\\
44& 297501075799& 490& 527& 1.08& 698& 1.42\\\
45& 303371455241& 500& 528& 1.06& 699& 1.40\\\
46& 304599508537& 514& 528& 1.03& 699& 1.36\\\
47& 416608695821& 516& 542& 1.05& 716& 1.39\\\
48& 461690510011& 532& 547& 1.03& 721& 1.36\\\
49& 614487453523& 534& 560& 1.05& 737& 1.38\\\
50& 738832927927& 540& 568& 1.05& 747& 1.38\\\
51& 1346294310749& 582& 596& 1.02& 780& 1.34\\\
52& 1408695493609& 588& 598& 1.02& 783& 1.33\\\
53& 1968188556461& 602& 614& 1.02& 801& 1.33\\\
54& 2614941710599& 652& 628& 0.96& 818& 1.25\\\
55& 7177162611713& 674& 678& 1.01& 876& 1.30\\\
56& 13829048559701& 716& 711& 0.99& 916& 1.28\\\
57& 19581334192423& 766& 729& 0.95& 937& 1.22\\\
58& 42842283925351& 778& 771& 0.99& 985& 1.27\\\
59& 90874329411493& 804& 812& 1.01& 1033& 1.28\\\
60& 171231342420521& 806& 847& 1.05& 1074& 1.33\\\
61& 218209405436543& 906& 861& 0.95& 1090& 1.20\\\
62& 1189459969825483& 916& 961& 1.05& 1205& 1.32\\\
63& 1686994940955803& 924& 982& 1.06& 1229& 1.33\\\
64& 1693182318746371& 1132& 982& 0.87& 1230& 1.09\\\
65& 43841547845541059& 1184& 1191& 1.01& 1468& 1.24\\\
66& 55350776431903243& 1198& 1207& 1.01& 1486& 1.24\\\
67& 80873624627234849& 1220& 1233& 1.01& 1516& 1.24\\\
68& 203986478517455989& 1224& 1297& 1.06& 1589& 1.30\\\
69& 218034721194214273& 1248& 1301& 1.04& 1594& 1.28\\\
70& 305405826521087869& 1272& 1325& 1.04& 1621& 1.27\\\
71& 352521223451364323& 1328& 1336& 1.01& 1632& 1.23\\\
72& 401429925999153707& 1356& 1345& 0.99& 1643& 1.21\\\
73& 418032645936712127& 1370& 1348& 0.98& 1646& 1.20\\\
74& 804212830686677669& 1442& 1395& 0.97& 1700& 1.18\\\
75& 1425172824437699411& 1476& 1437& 0.97& 1747& 1.18
\end{matrix}
A:Serial numbe, B:Natural number, C:$\max_{p_{n+1}\leqslant N}(p_{n+1}-p_{n})$, D:$logN(logN-2loglogN)+2$, E:$\frac{logN(logN-2loglogN)+2}{\max_{p_{n+1}\leqslant N}(p_{n+1}-p_{n})}$, F:$ (logN)^{2}$, G:$\frac{(logN)^{2}}{\max_{p_{n+1}\leqslant N}(p_{n+1}-p_{n})}$