Maximal gaps between prime k-tuples

© 2011-2013 by Alexei Kourbatov, JavaScripter.net/math.
An extended version of this article is now available:
Journal of Integer Sequences: Maximal gaps between prime k-tuples: a statistical approach, Article 13.5.2, MR3065331, arXiv:1301.2242.

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 log²p when p → ∞. Moreover, Shanks [8] stated that the maximal gaps are asymptotically equal to log²p. 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 log²p 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 average gaps between prime k-tuples near p are O(log^kp) as p → ∞, in agreement with the Hardy-Littlewood k-tuple conjecture [9,14]. Computations show that maximal gaps between k-tuples are at most about log p times the average gap, which implies that maximal gaps are O(log^k+1p) as p → ∞. Probability-based heuristics described below suggests a general formula predicting the size of maximal gaps between k-tuples: maximal gaps are approximately max(a, a(log(p/a) − b)), where a = C_klog^kp is the estimated average gap near p, and C_k and b are positive parameters depending on the type of k-tuple. This formula approximates maximal gaps better and in a wider range than a linear function of log^k+1p (cf. Rodriguez and Rivera [12] for the special case k = 2; note the issues caused by a linear approximation). In what follows, we will focus on three specific types of prime k-tuples:

twin primes (k = 2),

prime quadruplets (k = 4), and

prime sextuplets (k = 6).

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), quintuplets (k = 5), septuplets (k = 7), and decuplets (k = 10).

Definitions and notation

Twin primes are pairs of consecutive primes that have the form {p, p+2}. This is the densest repeatable pattern of two primes.

Prime quadruplets are clusters of four consecutive primes of the form {p, p+2, p+6, p+8}. This is the densest repeatable pattern of four primes.

Prime sextuplets are clusters of six consecutive primes of the form {p, p+4, p+6, p+10, p+12, p+16}. This is the densest repeatable pattern of six primes.

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; prime quadruplets are another specific type of prime k-tuples, with k = 4; and prime sextuplets are yet another type of prime k-tuples, with k = 6. (The densest 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(p). For example, the gap between the quadruplets {11,13,17,19} and {101,103,107,109} is g₄(101) = 90. The gap between the twin primes {17,19} and {29,31} is g₂(29) = 12.

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, log x denotes the natural logarithm of x, and log^kx = (log x)^k is log x raised to the k-th power.

Simple conjectures for the gap size

There are simple formulas for gaps between k-tuples of a specific type. These formulas give rough estimates of the gap g_k(p) that ends at a prime p:
(A) Average gaps between prime k-tuples are g_k(p) ≈ C_k log^kp ≡ a(p), where C_k are constants.
(B) Maximal gaps between prime k-tuples are O(log^k+1p): g_k(p) < M_k log^k+1p, where M_k ≈ C_k.
(C) Maximal gaps between prime k-tuples are asymptotically equal to C_klog^k+1p as p → ∞.

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,
C₂ ≈ 0.75739... (for twin primes)
C₃ ≈ 0.34986... (for prime triplets)
C₄ ≈ 0.240895... (for prime quadruplets)
C₆ ≈ 0.057808... (for prime sextuplets).

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. (For more data, see Appendixes as well as the following OEIS sequences: maximal gaps between triplets A201596, A201598, quintuplets A201062, A201073, septuplets A201051, A201251, and decuplets A202281 and A202361. Data in these sequences thus far seem to support our general conjectures – or at least do not contradict them. However, one would need to analyze huge amounts of additional data, especially for 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 (k ≥ 2) in (B) are less than 1; moreover, computations suggest that M_k ≈ C_k. In particular, for k = 2, 4, 6 statement (B) becomes:
Maximal gaps between 2-tuples (twin prime pairs) are O(log³p): g₂(p) < 0.76 log³p.
Maximal gaps between 4-tuples (prime quadruplets) are O(log⁵p): g₄(p) < 0.241 log⁵p.
Maximal gaps between 6-tuples (prime sextuplets) are O(log⁷p): g₆(p) < 0.058 log⁷p.

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 prime k-tuples are at most about log p times the average gap.

Can we predict the maximal gaps even better?
Probability-based heuristics can help!

Prime k-tuples are rare and seemingly random. Life offers many examples of unusually large intervals between rare random events: the longest runs of two-dice rolls without getting a twelve, maximal intervals between cars crossing a bridge at night, maximal intervals between clicks of a Geiger counter measuring very low radioactivity, etc. Schilling gives many more statistical examples of this kind in his paper The longest run of heads [15] and states “the log n law” for the length or duration of the longest runs. Reasoning as in [15], one can statistically estimate the expected maximal intervals between rare random events by expressing the maximal intervals in terms of the average intervals:

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

where a is the average interval between the rare events, and T is the total observation time or length (1 << a << T).

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(p) we define a heuristic maximal gap estimator E(max g_k(p)) – a function of p, with two more positive arguments, a and b:

E(max g_k(p)) = max(a, a log(p/a) − ba), with a = C_k log^kp.

Here, the role of the total observation time T is played by p (we are looking for maximal gaps that occur from 0 up to p). The role of average intervals a is played by the size of average gaps between k-tuples near p, i.e. we set a = C_k log^kp, as predicted by the Hardy-Littlewood k-tuple conjecture – see (A) above. The constants C_k are reciprocal to the Hardy-Littlewood constants, which are known to a high precision [9,11]. For instance, in case of twin primes (k = 2) we have a value reciprocal to the twin prime constant 2Π₂: C₂ = (2Π₂)⁻¹ = (1.3203236...)⁻¹ = 0.75739... Unlike simpler statistical problems with constant a, in our case the average gap a itself grows with p. (Using the Geiger counter analogy, this means that during a very long observation time T the radioactivity level not only is low, but gradually gets even lower, so the average intervals between clicks grow longer and longer.) The additional parameter b helps us compensate for this growth and achieve a better fit with known data on k-tuples. The choice b ≈ 2/k works well in most cases; the choice b ≈ 3 works even better for k = 1 (i.e. for maximal gaps between primes).

Is the prediction accurate?

Below are graphs and numerical data for k = 2, 4, 6. The diamond-shaped data points on the graphs are the actual maximal gaps between k-tuples obtained from a lengthy computation. We easily see that all maximal gaps g_k(p) are smaller than the empirical upper bound C_klog^k+1p (top red line). The estimator (blue curve) E(max g_k) = a(log(p/a) − b) approximates the actual maximal gaps quite closely (usually predicting the maximal gap sizes with accuracy ±2a or better). At first, the estimator curve seems to go way below the respective upper bound (red line). However, substituting a = C_k log^kp we can rewrite the estimator as

E(max g_k) = a log(p/a) − ab = a log p − a(log a + b) = C_klog^k+1p − o(log^k+1p),

which shows that E(max g_k) < C_klog^k+1p, but at the same time E(max g_k) is asymptotically equal to the upper bound C_klog^k+1p as p → ∞. This means that eventually the red line and blue curve will become “almost parallel” as p → ∞. That is to say, if the slope of the red line were even a little less than C_k – while the blue curve were to remain as it is – then the red line would intersect the blue curve at some (very distant) point.

Notes. Of course, it is not guaranteed that the formula a(log(p/a) − b) is valid for very big p. Importantly, for any p, the estimator does not predict the existence of a maximal gap g_k(p) anywhere near p – it only gives us a good guess for the size of g_k(p) if there is a maximal gap near p. As far as rigorous proofs are concerned, we don't even know whether there are infinitely many k-tuples of a given type − for example, whether there are infinitely many twin primes for k = 2. (The famous twin prime conjecture as well as the more general k-tuple conjecture thus far remain unproven. Thanks to the theorems of Zhang [21] and Maynard [22], we now know that at least some types of prime k-tuples do occur infinitely often.)

Maximal gaps between twin primes up to 1.2 × 10¹⁵

  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 g₂(p)/log³p is never greater than 0.76. (The maximal values of g₂(p) for p up to 1.2 × 10¹⁵ were first computed in 2008 by Richard Fischer [13] and later extended by Tomás Oliveira e Silva [24].)

Maximal gaps between prime quadruplets up to 10¹⁵:

    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         6983730

The ratio g₄(p)/log⁵p is never greater than 0.241.

Maximal gaps between prime 6-tuples up to 10¹⁵:

    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      1457965740

The ratio g₆(p)/log⁷p is never greater than 0.058; in fact, for all of the above gaps g₆(p), the ratio is far less.

On maximal gaps between primes

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

Maximal gaps are about a(log(p/a) − b), with a = log p and b ≈ 3.

This leads to the well-known Cramér and Shanks conjectures for prime gaps g(p):

lim sup(g(p)/log²p) = 1 as p → ∞ (Cramér [7]),

while maximal gaps between consecutive primes are asymptotically equal to log²p (Shanks [8]).

Indeed, for a = log p and any fixed b ≥ 0, we have log²p > a(log(p/a) − b) ~ log²p as 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 (log p). Based in part on this theorem, Granville [17] adjusted the Cramér-Shanks conjecture and proposed that, as p → ∞, lim sup(g(p)/log²p) ≥ 2e^−γ = 1.1229... This would mean that an infinite subsequence of maximal gaps must lie above the Cramér-Shanks upper limit log²p, i.e. above the red line on this graph – and this hypothetical subsequence (or an infinite subset thereof) must approach a line whose slope is about 1.1229 times steeper! However, for now, there is not a single data point for maximal prime gaps above log²p, where 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 sup G(x) / log²x = 1, while Shanks made the stronger conjecture that G(x) ~ log²x, but we are still a long way from proving these statements.
[Here G(x) denotes the largest prime gap below x.]

So do we possibly need a constant before log²x in the asymptotic equivalence G(x) ~ log²x? Or maybe the asymptotic equivalence is not valid at all? Maybe the best way we can hope to describe the maximal prime gaps is in weaker terms of an upper limit, lim sup (G(x)/log²x)? We'll live and see! (Let us hope to live long enough...)

Appendixes

The above heuristic argument appears to be applicable to prime k-tuples for all natural k, as long as the Hardy-Littlewood k-tuple conjecture [9,14] remains valid. However, we have purposely chosen only the cases k = 2, 4, 6 for the main presentation above. Data and specific conjectures for other values of k can be found in these appendixes:

Maximal gaps between prime triplets (k = 3)

Maximal gaps between prime quintuplets (k = 5)

Maximal gaps between prime septuplets (k = 7)

Maximal gaps between prime decuplets (k = 10)

Hardy-Littlewood constants and reciprocals

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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 4·10¹⁸, Math. Comp. 83, 2033-2060, 2014.

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