© 2011-2013 *by Alexei Kourbatov, JavaScripter.net/math*.

An extended version of this article is now available:

Gaps between consecutive primes have been extensively studied [1–8, 20–23].
The prime number theorem [6] suggests that “typical” prime gaps near *p* have the size about log *p*.
On the other hand, Cramér [7] conjectured that maximal gaps between primes near *p* are
*at most* about as large as log^{2}*p* when *p* → ∞^{2}*p*.
All maximal gaps between primes are now known, up to low 19-digit primes [1–5, 23].
This data apparently supports the Cramér and Shanks conjectures:
thus far, if we divide by log^{2}*p* the prime gap ending at *p*, the resulting ratio is always less than one
– but tends to grow closer to one, albeit very slowly and irregularly.

Less is known about maximal gaps between *prime constellations*,
or *prime k-tuples* [9–14, 19, 24].
We can conjecture that

The observations can be generalized to other *k*-tuples;
however, the numeric constants will change depending on the specific type of *k*-tuple.
See Appendixes for data on maximal gaps between prime
triplets (*k* = 3*k* = 5*k* = 7*k* = 10

*Twin primes* are pairs of consecutive primes that have the form *p*, *p*+2}

*Prime quadruplets* are clusters of four consecutive primes of the form *p*, *p*+2, *p*+6, *p*+8}

*Prime sextuplets* are clusters of six consecutive primes of the form
*p*, *p*+4, *p*+6, *p*+10, *p*+12, *p*+16}

*Prime k-tuples* are clusters of *k* consecutive primes that have a repeatable pattern.
Thus, twin primes are a specific type of prime *k*-tuples, with *k* = 2;*k*-tuples, with *k* = 4;*k*-tuples, with *k* = 6.*k*-tuples possible for a given *k* may also be called
prime constellations or
prime *k*-tuplets.)

*Gaps* between prime *k*-tuples are center-to-center distances between consecutive *k*-tuples.
If the prime at the “far” end of the gap is *p*, we denote the gap *g _{k}*(

A *maximal gap* is a gap that is strictly greater than all preceding gaps. In other words,
a maximal gap is the first occurrence of a gap *at least this size.*
As an example, consider gaps between prime quadruplets (4-tuples):
the gap of 90 preceding the quadruplet {101,103,107,109} is a maximal gap
(i.e. the first occurrence of a gap of at least 90),
while the gap of 90 preceding {191,193,197,199} is *not* a maximal gap
(*not* the first occurrence of a gap at least this size).
A synonym for *maximal gap* is *record gap*; see e.g. OEIS Sequences
A113274,
A113404,
A200503
(record gaps between twins, quadruplets, and sextuplets, respectively).

Hereafter *p* denotes a prime, *x**natural logarithm* of *x*, and
^{k}*x* = (log *x*)^{k}*x**k*-th power.

(A)

(B)

(C)

Statement (A) follows from the
Hardy-Littlewood *k*-tuple conjecture [9,14]
that predicts the total counts of prime *k*-tuples (and thereby *average frequencies* of *k*-tuples);
the constants *C _{k}* are reciprocals of the
Hardy-Littlewood constants.
For example,

Statements (B) and (C) are also conjectural – they are suggested by heuristics and supported by
large sets of known maximal gaps such as listed below for *k* = 2, 4, 6.*k*-tuples with larger *k*, to boost the confidence level.
Of course, no amount of data can replace a formal proof – which we do not have at this time,
much like we still lack a formal proof of similar conjectures for primes [7,8].)

The constants *M _{k}* (

Statement (C) is stronger than (B) and analogous to the Shanks conjecture for primes [8]. Together, statements (A), (B), (C) tell us that:

Maximal gaps between primek-tuples are at most aboutlog ptimes the average gap.

Probability-based heuristics can help!

whereE(max interval) =alog(T/a) +O(a),with standard deviation of about a,

For lack of a better model,
we will simulate the distribution of prime *k*-tuples by a timeline of rare random events.
(A quick nod to purists: Yes, we have to remember that consecutive prime *k*-tuples *are not independent events*;
thus they are beyond the proven range of applicability of most statistical models. The same is true for consecutive primes.
But we are *not proving a theorem* here – just stating conjectures.)
By analogy with the above statistical formula, for maximal gaps *g _{k}*(

Here, the role of the total observation timeE(maxg(_{k}p)) = max(a,alog(p/a) −ba),with a=Clog_{k}^{k}p.

Below are graphs and numerical data for *k* = 2, 4, 6.*k*-tuples obtained from a lengthy computation.
We easily see that all maximal gaps *g _{k}*(

which shows thatE(maxg) =_{k}alog(p/a) −ab=alogp−a(loga+b) =Clog_{k}^{k+1}p−o(log^{k+1}p),

**Notes.**
Of course, it is not guaranteed that the formula *a*(log(*p*/*a*) − *b*)*p*.
Importantly, for any *p*, the estimator does *not* predict the *existence* of
a maximal gap *g _{k}*(

1st twin pair: 2nd twin pair: Gap g2(p): 3 5 2 5 11 6 17 29 12 41 59 18 71 101 30 311 347 36 347 419 72 659 809 150 2381 2549 168 5879 6089 210 13397 13679 282 18539 18911 372 24419 24917 498 62297 62927 630 187907 188831 924 687521 688451 930 688451 689459 1008 850349 851801 1452 2868959 2870471 1512 4869911 4871441 1530 9923987 9925709 1722 14656517 14658419 1902 17382479 17384669 2190 30752231 30754487 2256 32822369 32825201 2832 96894041 96896909 2868 136283429 136286441 3012 234966929 234970031 3102 248641037 248644217 3180 255949949 255953429 3480 390817727 390821531 3804 698542487 698547257 4770 2466641069 2466646361 5292 4289385521 4289391551 6030 19181736269 19181742551 6282 24215097497 24215103971 6474 24857578817 24857585369 6552 40253418059 40253424707 6648 42441715487 42441722537 7050 43725662621 43725670601 7980 65095731749 65095739789 8040 134037421667 134037430661 8994 198311685749 198311695061 9312 223093059731 223093069049 9318 353503437239 353503447439 10200 484797803249 484797813587 10338 638432376191 638432386859 10668 784468515221 784468525931 10710 794623899269 794623910657 11388 1246446371789 1246446383771 11982 1344856591289 1344856603427 12138 1496875686461 1496875698749 12288 2156652267611 2156652280241 12630 2435613754109 2435613767159 13050 4491437003327 4491437017589 14262 13104143169251 13104143183687 14436 14437327538267 14437327553219 14952 18306891187511 18306891202907 15396 18853633225211 18853633240931 15720 23275487664899 23275487681261 16362 23634280586867 23634280603289 16422 38533601831027 38533601847617 16590 43697538391391 43697538408287 16896 56484333976919 56484333994001 17082 74668675816277 74668675834661 18384 116741875898981 116741875918727 19746 136391104728629 136391104748621 19992 221346439666109 221346439686641 20532 353971046703347 353971046725277 21930 450811253543219 450811253565767 22548 742914612256169 742914612279527 23358 1121784847637957 1121784847661339 23382 1149418981410179 1149418981435409 25230

The ratio *g*_{2}(*p*)/log^{3}*p* is never greater than 0.76.
(The maximal values of *g*_{2}(*p*)*p* up to ^{15}

1st 4-tuple: 2nd 4-tuple: Gap g4(p): 5 11 6 11 101 90 191 821 630 821 1481 660 2081 3251 1170 3461 5651 2190 5651 9431 3780 25301 31721 6420 34841 43781 8940 88811 97841 9030 122201 135461 13260 171161 187631 16470 301991 326141 24150 739391 768191 28800 1410971 1440581 29610 1468631 1508621 39990 2990831 3047411 56580 3741161 3798071 56910 5074871 5146481 71610 5527001 5610461 83460 8926451 9020981 94530 17186591 17301041 114450 21872441 22030271 157830 47615831 47774891 159060 66714671 66885851 171180 76384661 76562021 177360 87607361 87797861 190500 122033201 122231111 197910 132574061 132842111 268050 204335771 204651611 315840 628246181 628641701 395520 1749443741 1749878981 435240 2115383651 2115824561 440910 2128346411 2128859981 513570 2625166541 2625702551 536010 2932936421 2933475731 539310 3043111031 3043668371 557340 3593321651 3593956781 635130 5675642501 5676488561 846060 25346635661 25347516191 880530 27329170151 27330084401 914250 35643379901 35644302761 922860 56390149631 56391153821 1004190 60368686121 60369756611 1070490 71335575131 71336662541 1087410 76427973101 76429066451 1093350 87995596391 87996794651 1198260 96616771961 96618108401 1336440 151023350501 151024686971 1336470 164550390671 164551739111 1348440 171577885181 171579255431 1370250 210999769991 211001269931 1499940 260522319641 260523870281 1550640 342611795411 342614346161 2550750 1970587668521 1970590230311 2561790 4231588103921 4231591019861 2915940 5314235268731 5314238192771 2924040 7002440794001 7002443749661 2955660 8547351574961 8547354997451 3422490 15114108020021 15114111476741 3456720 16837633318811 16837637203481 3884670 30709975578251 30709979806601 4228350 43785651890171 43785656428091 4537920 47998980412211 47998985015621 4603410 55341128536691 55341133421591 4884900 92944027480721 92944033332041 5851320 412724560672211 412724567171921 6499710 473020890377921 473020896922661 6544740 885441677887301 885441684455891 6568590 947465687782631 947465694532961 6750330 979876637827721 979876644811451 6983730The ratio

1st 6-tuple: 2nd 6-tuple: Gap g6(p): 7 97 90 97 16057 15960 19417 43777 24360 43777 1091257 1047480 3400207 6005887 2605680 11664547 14520547 2856000 37055647 40660717 3605070 82984537 87423097 4438560 89483827 94752727 5268900 94752727 112710877 17958150 381674467 403629757 21955290 1569747997 1593658597 23910600 2019957337 2057241997 37284660 5892947647 5933145847 40198200 6797589427 6860027887 62438460 14048370097 14112464617 64094520 23438578897 23504713147 66134250 24649559647 24720149677 70590030 29637700987 29715350377 77649390 29869155847 29952516817 83360970 45555183127 45645253597 90070470 52993564567 53086708387 93143820 58430706067 58528934197 98228130 93378527647 93495691687 117164040 97236244657 97367556817 131312160 240065351077 240216429907 151078830 413974098817 414129003637 154904820 419322931117 419481585697 158654580 422088931207 422248594837 159663630 427190088877 427372467157 182378280 610418426197 610613084437 194658240 659829553837 660044815597 215261760 660863670277 661094353807 230683530 853633486957 853878823867 245336910 1089611097007 1089869218717 258121710 1247852774797 1248116512537 263737740 1475007144967 1475318162947 311017980 1914335271127 1914657823357 322552230 1953892356667 1954234803877 342447210 3428196061177 3428617938787 421877610 9367921374937 9368397372277 475997340 10254799647007 10255307592697 507945690 13786576306957 13787085608827 509301870 21016714812547 21017344353277 629540730 33157788914347 33158448531067 659616720 41348577354307 41349374379487 797025180 72702520226377 72703333384387 813158010 89165783669857 89166606828697 823158840 122421000846367 122421855415957 854569590 139864197232927 139865086163977 888931050 147693859139077 147694869231727 1010092650 186009633998047 186010652137897 1018139850 202607131405027 202608270995227 1139590200 332396845335547 332397997564807 1152229260 424681656944257 424682861904937 1204960680 437804272277497 437805730243237 1457965740The ratio

For comparison, here is the same model applied to maximal gaps between primes (A005250):

Maximal gaps are aboutThis leads to the well-known Cramér and Shanks conjectures for prime gapswith a(log(p/a) −b),and a= logp. b≈ 3

whilelim sup( ,g(p)/log^{2}p) = 1 asp→ ∞ (Cramér [7])

Indeed, for *a* = log *p**any* fixed *b* ≥ 0^{2}*p* > *a*(log(*p*/*a*) − *b*) `~`

log^{2}*p**p* → ∞

**An adjustment to the Cramér-Shanks conjecture?**
Maier's theorem [16] states that
there are (relatively short) intervals where typical gaps between primes are greater than the average *p*)*p* → ∞*g*(*p*)/log^{2}*p*) ≥ 2*e*^{−γ} = 1.1229...*above*
the Cramér-Shanks upper limit ^{2}*p**there is not a single data point* for maximal prime gaps above ^{2}*p**p* denotes the end-of-gap prime.
Apparently Helmut Maier himself did not feel that the Cramér-Shanks conjecture was necessarily in danger because of his theorem;
thus, Maier and Pomerance [18] simply remarked in 1990 (five years after the publication of Maier's theorem):

Cramér conjectured that lim supSo do we possibly need a constant before logG(x) / log^{2}x= 1, while Shanks made the stronger conjecture that, but we are still a long way from proving these statements. G(x)`~`

log^{2}x

[Heredenotes the largest prime gap below G(x)x.]

`~`

log1. Young, J. and Potler, A. First Occurrence Prime Gaps. Math. Comp. 52, 221-224, 1989.

2. Nicely, Thomas R. List of prime gaps. http://www.trnicely.net/gaps/gaplist.html (2011)

3. Oliveira e Silva, Tomás. Gaps between consecutive primes. http://www.ieeta.pt/~tos/gaps.html (2001-2013)

4. Ribenboim, Paulo. The little book of big primes, New York, Springer-Verlag, 1991.

(Second edition: The little book of bigger primes, New York, Springer-Verlag, 2004)

5. Weisstein, Eric W. Prime Gaps.
From MathWorld – A Wolfram Web Resource.

http://mathworld.wolfram.com/PrimeGaps.html (2011)

6. Hardy, Godfrey H. and Wright, Edward M. An Introduction to the Theory of Numbers, 6th ed. Oxford, Oxford University Press, 2008, p.10.

7. Cramér, Harald. On the Order of Magnitude of the Difference between Consecutive Prime Numbers. Acta Arith. 2, 23-46, 1936.

8. Shanks, Daniel. On Maximal Gaps Between Successive Primes. Math. Comp. 18, 646-651, 1964.

9. Hardy, Godfrey H. and Littlewood, John E.
Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes.
Acta Math. 44, 1-70, 1923.

http://www.springerlink.com/index/D700T434065K0230.pdf

10. Smith, Herschel F. On a generalization of the prime pair problem,
Mathematical Tables and Other Aids to Computation, v. 11, 1957, p. 274.

http://www.ams.org/journals/mcom/1957-11-060/S0025-5718-1957-0094314-8/home.html

11. Forbes, Anthony D. Prime *k*-tuplets.
http://anthony.d.forbes.googlepages.com/ktuplets.htm (2011)

12. Rivera, Carlos and Rodriguez, Luis. Gaps between consecutive twin prime pairs

http://www.primepuzzles.net/conjectures/conj_066.htm

13. Fischer, Richard. Maximale Lücken (Intervallen) von Primzahlenzwillingen.

http://www.fermatquotient.com/PrimLuecken/ZwillingsRekordLuecken (2008)

14. Weisstein, Eric W. k-Tuple Conjecture.
From MathWorld – A Wolfram Web Resource.

http://mathworld.wolfram.com/k-TupleConjecture.html (2011)

15. Schilling, Mark F. The longest run of heads. The College Mathematics Journal, v.21, No.3, 196-207, 1990.

16. Maier, Helmut. Primes in short intervals, Michigan Math. J., v.32, 221-225, 1985.

17. Granville, Andrew. Harald Cramér and the distribution of prime numbers.
Scand. Act. J. 1, 12-28, 1995.

http://www.dms.umontreal.ca/~andrew/PDF/cramer.pdf

18. Maier, Helmut and Pomerance, Carl. Unusually large gaps between consecutive primes. Transactions of the AMS, v.322, No. 1, 201-237, 1990.

19. Wolf, Marek. Some remarks on the distribution of twin primes, http://arxiv.org/abs/math/0105211 (2001)

20. Wolf, Marek. Some heuristics on the gaps between consecutive primes,
http://arxiv.org/abs/1102.0481 (2011)

**2013 — New Proofs**

21. Zhang, Yitang. Bounded gaps between primes, Annals of Math. 179, No.3, 1121-1174, 2014.

22. Maynard, James. Small gaps between primes, http://arxiv.org/abs/1311.4600 (2013)

**Recent Computations and Heuristics**

23. Oliveira e Silva, Tomás, Herzog, Siegfried, and Pardi, Silvio.
Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10^{18},

24. Oliveira e Silva, Tomás, Gaps between twin primes. http://sweet.ua.pt/tos/twin_gaps.html (2013)

25. Kourbatov, Alexei, Maximal gaps between prime k-tuples: a statistical approach. Journal of Integer Sequences, 16, Article 13.5.2, 2013. arXiv:1301.2242

26. Kourbatov, Alexei, Tables of record gaps between prime constellations, arXiv:1309.4053 (2013)

27. Kourbatov, Alexei, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, Int. Journal of Statistics and Probability, 3, No.2, 18-29, 2014. arXiv:1401.6959

Copyright © 2011-2014, Alexei Kourbatov, JavaScripter.net.