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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 $$\lim_{n\rightarrow \infty }\sup\frac{p_{n+1}-p_{n}}{(\log p_{n})^{2}}=1$$which is still an unproven conjecture.

I found a conjecture about the prime maximal gaps that $$\max_{p_{n+1}\leqslant N }(p_{n+1}-p_{n})\approx \log N(\log N-2\log\log N)+2$$ 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?


\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 number, B:Natural number, C:$\max_{p_{n+1}\leqslant N}(p_{n+1}-p_{n})$, D:$\log N(\log N-2\log\log N)+2$, E:$\frac{\log N(\log N-2\log\log N)+2}{\max_{p_{n+1}\leqslant N}(p_{n+1}-p_{n})}$, F:$ (\log N)^{2}$, G:$\frac{(\log N)^{2}}{\max_{p_{n+1}\leqslant N}(p_{n+1}-p_{n})}$

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To me this looks a lot stronger than Legendre's, Andrica's or Oppermann's conjecture. Prove one of those first. –  Hagen von Eitzen Jul 20 '13 at 11:59
1  
Posted about a month ago to MO, mathoverflow.net/questions/134315/… --- YOU should have posted this link! –  Gerry Myerson Jul 20 '13 at 12:42
    
@Gerry Myerson thank you very much. –  Wenlong Du Jul 20 '13 at 13:12

1 Answer 1

Strictly interpreted your result would imply Cramér's conjecture, which as you already said is still open, so a proof is not likely to be forthcoming.

But in "Some Conjectures on the Gaps between Consecutive Primes" Wolf gives a heuristic argument to support a similar conjecture that $$ \begin{align} \max_{p_{n+1}\le N }(p_{n+1}-p_{n}) &\approx \frac{N}{\pi(N)}(2\log \pi(N) -\log N+\log 2C_2) \\ &\approx \log N(\log N-2\log\log N+\log 2C_2) \end{align} $$ where $\pi(N)$ is the prime counting function and $C_2=0.660\cdots$ is the twin primes constant.

However, Wolf cites Granville "Unexpected Irregularities in the Distribution of Prime Numbers" in which he suggests that the proof of Maier's theorem might be adapted to show that there are infinitely many primes with $p_{n+1}-p_n>2e^{-\gamma}\log^2 p_n$, which would disprove Cramér's conjecture, and would say that your estimate and Wolf's could at best be close "almost always."

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Should that formula have $2 \log \pi(n)$ instead of $\log \pi(n)$? That is needed to make the next line come out. –  marty cohen Jul 22 '13 at 3:05
    
@martycohen Indeed you are right. –  Zander Jul 22 '13 at 11:53

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