Prime gap

Prime gap frequency distribution for primes up to 1.6 billion. Peaks occur at multiples of 6.[1]

A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n + 1)-th and the n-th prime numbers, i.e.

${\displaystyle g_{n}=p_{n+1}-p_{n}.\ }$

We have g1 = 1, g2 = g3 = 2, and g4 = 4. The sequence (gn) of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered.

The first 60 prime gaps are:

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... (sequence A001223 in the OEIS).

By the definition of gn every prime can be written as

${\displaystyle p_{n+1}=2+\sum _{i=1}^{n}g_{i}.}$

Simple observations

The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps g2 and g3 between the primes 3, 5, and 7.

For any integer n, the factorial n! is the product of all positive integers up to and including n. Then in the sequence

${\displaystyle 2!+2,3!+3,\ldots ,n!+n}$

the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of n − 1 consecutive composite integers, and it must belong to a gap between primes having length at least n. It follows that there are gaps between primes that are arbitrarily large, that is, for any integer N, there is an integer m with gmN.

However, prime gaps of n numbers can occur at numbers much smaller than n!. For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000.

The average gap between primes increases as the natural logarithm of the integer, and therefore the ratio of the prime gap to the integers involved decreases (and is asymptotically zero). This is a consequence of the prime number theorem. From a heuristic view, we expect the probability that the ratio of the length of the gap to the natural logarithm is greater than or equal to a fixed positive number k to be ek; consequently the ratio can be arbitrarily large. Indeed, the ratio of the gap to the number of digits of the integers involved does increase without bound. This is a consequence of a result by Westzynthius.[2]

In the opposite direction, the twin prime conjecture posits that gn = 2 for infinitely many integers n.

Numerical results

Usually the ratio of gn / ln(pn) is called the merit of the gap gn . As of September 2017, the largest known prime gap with identified probable prime gap ends has length 6582144, with 216841-digit probable primes found by Martin Raab.[3] This gap has merit M = 13.1829. The largest known prime gap with identified proven primes as gap ends has length 1113106 and merit 25.90, with 18662-digit primes found by P. Cami, M. Jansen and J. K. Andersen.[4][5]

As of December 2017, the largest known merit value and first with merit over 40, as discovered by the Gapcoin network, is 41.93878373 with the 87-digit prime 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227. The prime gap between it and the next prime is 8350.[6]

Largest known merit values (as of January 2018)[6][7][8]
Merit gn digits pn Date Discoverer
41.938784 8350 87 See above 2017 Gapcoin
39.620154 15900 175 3483347771 × 409#/30 − 7016 2017 Dana Jacobsen
38.066960 18306 209 650094367 × 491#/2310 − 8936 2017 Dana Jacobsen
37.824126 8382 97 512950801 × 229#/5610 − 4138 2018 Dana Jacobsen
37.005294 26054 306 1780005161 × 719#/30 − 17768 2017 Dana Jacobsen

The Cramér–Shanks–Granville ratio is the ratio of gn / (ln(pn))2.[6] If we discard anomalously high values of the ratio for the primes 2, 3, 7, then the greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. Other record terms can be found at .

We say that gn is a maximal gap, if gm < gn for all m < n. As of August 2018 the largest known maximal prime gap has length 1550, found by Bertil Nyman. It is the 80th maximal gap, and it occurs after the prime 18361375334787046697.[9] Other record (maximal) gap sizes can be found in , with the corresponding primes pn in , and the values of n in .

The 80 known maximal prime gaps
Number 1 to 27
# gn pn n
1 1 2 1
2 2 3 2
3 4 7 4
4 6 23 9
5 8 89 24
6 14 113 30
7 18 523 99
8 20 887 154
9 22 1,129 189
10 34 1,327 217
11 36 9,551 1,183
12 44 15,683 1,831
13 52 19,609 2,225
14 72 31,397 3,385
15 86 155,921 14,357
16 96 360,653 30,802
17 112 370,261 31,545
18 114 492,113 40,933
19 118 1,349,533 103,520
20 132 1,357,201 104,071
21 148 2,010,733 149,689
22 154 4,652,353 325,852
23 180 17,051,707 1,094,421
24 210 20,831,323 1,319,945
25 220 47,326,693 2,850,174
26 222 122,164,747 6,957,876
27 234 189,695,659 10,539,432
Number 28 to 54
# gn pn n
28 248 191,912,783 10,655,462
29 250 387,096,133 20,684,332
30 282 436,273,009 23,163,298
31 288 1,294,268,491 64,955,634
32 292 1,453,168,141 72,507,380
33 320 2,300,942,549 112,228,683
34 336 3,842,610,773 182,837,804
35 354 4,302,407,359 203,615,628
36 382 10,726,904,659 486,570,087
37 384 20,678,048,297 910,774,004
38 394 22,367,084,959 981,765,347
39 456 25,056,082,087 1,094,330,259
40 464 42,652,618,343 1,820,471,368
41 468 127,976,334,671 5,217,031,687
42 474 182,226,896,239 7,322,882,472
43 486 241,160,624,143 9,583,057,667
44 490 297,501,075,799 11,723,859,927
45 500 303,371,455,241 11,945,986,786
46 514 304,599,508,537 11,992,433,550
47 516 416,608,695,821 16,202,238,656
48 532 461,690,510,011 17,883,926,781
49 534 614,487,453,523 23,541,455,083
50 540 738,832,927,927 28,106,444,830
51 582 1,346,294,310,749 50,070,452,577
52 588 1,408,695,493,609 52,302,956,123
53 602 1,968,188,556,461 72,178,455,400
54 652 2,614,941,710,599 94,906,079,600
Number 55 to 80
# gn pn n
55 674 7,177,162,611,713 251,265,078,335
56 716 13,829,048,559,701 473,258,870,471
57 766 19,581,334,192,423 662,221,289,043
58 778 42,842,283,925,351 1,411,461,642,343
59 804 90,874,329,411,493 2,921,439,731,020
60 806 171,231,342,420,521 5,394,763,455,325
61 906 218,209,405,436,543 6,822,667,965,940
62 916 1,189,459,969,825,483 35,315,870,460,455
63 924 1,686,994,940,955,803 49,573,167,413,483
64 1,132 1,693,182,318,746,371 49,749,629,143,526
65 1,184 43,841,547,845,541,059 1,175,661,926,421,598
66 1,198 55,350,776,431,903,243 1,475,067,052,906,945
67 1,220 80,873,624,627,234,849 2,133,658,100,875,638
68 1,224 203,986,478,517,455,989 5,253,374,014,230,870
69 1,248 218,034,721,194,214,273 5,605,544,222,945,291
70 1,272 305,405,826,521,087,869 7,784,313,111,002,702
71 1,328 352,521,223,451,364,323 8,952,449,214,971,382
72 1,356 401,429,925,999,153,707 10,160,960,128,667,332
73 1,370 418,032,645,936,712,127 10,570,355,884,548,334
74 1,442 804,212,830,686,677,669 20,004,097,201,301,079
75 1,476 1,425,172,824,437,699,411 34,952,141,021,660,495
76 1,488 5,733,241,593,241,196,731 135,962,332,505,694,894
77 1,510 6,787,988,999,657,777,797 160,332,893,561,542,066
78 1,526 15,570,628,755,536,096,243 360,701,908,268,316,580
79 1,530 17,678,654,157,568,189,057 408,333,670,434,942,092
80 1,550 18,361,375,334,787,046,697 423,731,791,997,205,041

Further results

Upper bounds

Bertrand's postulate, proven in 1852, states that there is always a prime number between k and 2k, so in particular pn+1 < 2pn, which means gn < pn.

The prime number theorem, proven in 1896, says that the average length of the gap between a prime p and the next prime will asymptotically approach ln(p) for sufficiently large primes. The actual length of the gap might be much more or less than this. However, one can deduce from the prime number theorem an upper bound on the length of prime gaps:

For every ${\displaystyle \epsilon >0}$, there is a number ${\displaystyle N}$ such that for all ${\displaystyle n>N}$

${\displaystyle g_{n}.

One can also deduce that the gaps get arbitrarily smaller in proportion to the primes: the quotient

${\displaystyle \lim _{n\to \infty }{\frac {g_{n}}{p_{n}}}=0.}$

Hoheisel (1930) was the first to show[10] that there exists a constant θ < 1 such that

${\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log(x)}}{\text{ as }}x\to \infty ,}$

hence showing that

${\displaystyle g_{n}

for sufficiently large n.

Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,[11] and to θ = 3/4 + ε, for any ε > 0, by Chudakov.[12]

A major improvement is due to Ingham,[13] who showed that for some positive constant c, if

${\displaystyle \zeta (1/2+it)=O(t^{c})\,}$ then ${\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log(x)}}}$ for any ${\displaystyle \theta >(1+4c)/(2+4c).}$

Here, O refers to the big O notation, ζ denotes the Riemann zeta function and π the prime-counting function. Knowing that any c > 1/6 is admissible, one obtains that θ may be any number greater than 5/8.

An immediate consequence of Ingham's result is that there is always a prime number between n3 and (n + 1)3, if n is sufficiently large.[14] The Lindelöf hypothesis would imply that Ingham's formula holds for c any positive number: but even this would not be enough to imply that there is a prime number between n2 and (n + 1)2 for n sufficiently large (see Legendre's conjecture). To verify this, a stronger result such as Cramér's conjecture would be needed.

Huxley in 1972 showed that one may choose θ = 7/12 = 0.58(3).[15]

A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.[16]

In 2005, Daniel Goldston, János Pintz and Cem Yıldırım proved that

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0}$

and 2 years later improved this[17] to

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty .}$

In 2013, Yitang Zhang proved that

${\displaystyle \liminf _{n\to \infty }g_{n}<7\cdot 10^{7},}$

meaning that there are infinitely many gaps that do not exceed 70 million.[18] A Polymath Project collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013.[19] In November 2013, James Maynard introduced a new refinement of the GPY sieve, allowing him to reduce the bound to 600 and show that for any m there exists a bounded interval with an infinite number of translations each of which containing m prime numbers.[20] Using Maynard's ideas, the Polymath project improved the bound to 246;[19][21] assuming the Elliott–Halberstam conjecture and its generalized form, N has been reduced to 12 and 6, respectively.[19]

Lower bounds

In 1931, Erik Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,[2]

${\displaystyle \limsup _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=\infty .}$

In 1938, Robert Rankin proved the existence of a constant c > 0 such that the inequality

${\displaystyle g_{n}>{\frac {c\log n\log \log n\log \log \log \log n}{(\log \log \log n)^{2}}}}$

holds for infinitely many values n, improving the results of Westzynthius and Paul Erdős. He later showed that one can take any constant c < eγ, where γ is the Euler–Mascheroni constant. The value of the constant c was improved in 1997 to any value less than 2eγ.[22]

Paul Erdős offered a \$10,000 prize for a proof or disproof that the constant c in the above inequality may be taken arbitrarily large.[23] This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.[24][25]

The result was further improved to

${\displaystyle g_{n}>{\frac {c\log n\log \log n\log \log \log \log n}{\log \log \log n}}}$

for infinitely many values of n by Ford–Green–Konyagin–Maynard–Tao.[26]

In the spirit of Erdős' original prize, Terence Tao offered 10,000 USD for a proof that c may be taken arbitrarily large in this inequality.[27]

Lower bounds for chains of primes have also been determined.[28]

Prime gap function

Even better results are possible under the Riemann hypothesis. Harald Cramér proved[29] that the Riemann hypothesis implies the gap gn satisfies

${\displaystyle g_{n}=O({\sqrt {p_{n}}}\log p_{n}),}$

using the big O notation. (In fact this result needs only the weaker Lindelöf hypothesis, if you can tolerate an infinitesimally smaller exponent.[30]) Later, he conjectured that the gaps are even smaller. Roughly speaking, Cramér's conjecture states that

${\displaystyle g_{n}=O\left((\log p_{n})^{2}\right).}$

Firoozbakht's conjecture states that ${\displaystyle p_{n}^{1/n}}$ (where ${\displaystyle p_{n}}$ is the nth prime) is a strictly decreasing function of n, i.e.,

${\displaystyle p_{n+1}^{1/(n+1)}

If this conjecture is true, then the function ${\displaystyle g_{n}=p_{n+1}-p_{n}}$ satisfies ${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}{\text{ for all }}n>4.}$[31] It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz[32][33][34] which suggest that ${\displaystyle g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}}$ infinitely often for any ${\displaystyle \varepsilon >0,}$ where ${\displaystyle \gamma }$ denotes the Euler–Mascheroni constant.

Meanwhile, Oppermann's conjecture is weaker than Cramér's conjecture. The expected gap size with Oppermann's conjecture is on the order of

${\displaystyle g_{n}<{\sqrt {p_{n}}}.}$

As a result, under Oppermann's conjecture – there exists ${\displaystyle m}$ (probably ${\displaystyle m=30}$) for which every natural ${\displaystyle n>m}$ satisfies ${\displaystyle g_{n}<{\sqrt {p_{n}}}.}$

Andrica's conjecture, which is a weaker conjecture than Oppermann's, states that[35]

${\displaystyle g_{n}<2{\sqrt {p_{n}}}+1.}$

This is a slight strengthening of Legendre's conjecture that between successive square numbers there is always a prime.

Polignac's conjecture states that every positive even number k occurs as a prime gap infinitely often. The case k = 2 is the twin prime conjecture. The conjecture has not yet been proven or disproven for any specific value of k, but Zhang Yitang's result proves that it is true for at least one (currently unknown) value of k which is smaller than 70,000,000; as discussed above, this upper bound has been improved to 246.

As an arithmetic function

The gap gn between the nth and (n + 1)st prime numbers is an example of an arithmetic function. In this context it is usually denoted dn and called the prime difference function.[35] The function is neither multiplicative nor additive.

References

1. ^ "Hidden structure in the randomness of the prime number sequence?", S. Ares & M. Castro, 2005
2. ^ a b Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind", Commentationes Physico-Mathematicae Helsingsfors (in German), 5: 1–37, JFM 57.0186.02, Zbl 0003.24601.
3. ^
4. ^ Andersen, Jens Kruse. "The Top-20 Prime Gaps". Retrieved 2014-06-13.
5. ^ A proven prime gap of 1113106
6. ^ a b c NEW PRIME GAP OF MAXIMUM KNOWN MERIT
7. ^ Dynamic prime gap statistics
8. ^ TABLES OF PRIME GAPS
9. ^ NEW MAXIMAL PRIME GAPS OF 1530 AND 1550
10. ^ Hoheisel, G. (1930). "Primzahlprobleme in der Analysis". Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. 33: 3–11. JFM 56.0172.02.
11. ^ Heilbronn, H. A. (1933). "Über den Primzahlsatz von Herrn Hoheisel". Mathematische Zeitschrift. 36 (1): 394–423. doi:10.1007/BF01188631. S2CID 123216472.
12. ^ Tchudakoff, N. G. (1936). "On the difference between two neighboring prime numbers". Mat. Sb. 1: 799–814.
13. ^ Ingham, A. E. (1937). "On the difference between consecutive primes". Quarterly Journal of Mathematics. Oxford Series. 8 (1): 255–266. Bibcode:1937QJMat...8..255I. doi:10.1093/qmath/os-8.1.255.
14. ^ Cheng, Yuan-You Fu-Rui (2010). "Explicit estimate on primes between consecutive cubes". Rocky Mt. J. Math. 40: 117–153. arXiv:0810.2113. doi:10.1216/rmj-2010-40-1-117. S2CID 15502941. Zbl 1201.11111.
15. ^ Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones Mathematicae. 15 (2): 164–170. Bibcode:1971InMat..15..164H. doi:10.1007/BF01418933. S2CID 121217000.
16. ^ Baker, R. C.; Harman, G.; Pintz, J. (2001). "The difference between consecutive primes, II". Proceedings of the London Mathematical Society. 83 (3): 532–562. doi:10.1112/plms/83.3.532.
17. ^ Goldston, D. A.; Pintz, J.; Yildirim, C. Y. (2007). "Primes in Tuples II". arXiv:0710.2728 [math.NT].
18. ^ Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761.
19. ^ a b c "Bounded gaps between primes". Polymath. Retrieved 2013-07-21.
20. ^ Maynard, James (2015). "Small gaps between primes". Annals of Mathematics. 181 (1): 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7. MR 3272929. S2CID 55175056.
21. ^ D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences. 1 (12). arXiv:1407.4897. doi:10.1186/s40687-014-0012-7. MR 3373710. S2CID 119699189.
22. ^ Pintz, J. (1997). "Very large gaps between consecutive primes". J. Number Theory. 63 (2): 286–301. doi:10.1006/jnth.1997.2081.
23. ^ Erdős, Paul; Bollobás, Béla; Thomason, Andrew, eds. (1997). Combinatorics, Geometry and Probability: A Tribute to Paul Erdös. Cambridge University Press. p. 1. ISBN 9780521584722.
24. ^ Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence (2016). "Large gaps between consecutive prime numbers". Ann. of Math. 183 (3): 935–974. arXiv:1408.4505. doi:10.4007/annals.2016.183.3.4. MR 3488740. S2CID 16336889.
25. ^ Maynard, James (2016). "Large gaps between primes". Ann. of Math. 183 (3): 915–933. arXiv:1408.5110. doi:10.4007/annals.2016.183.3.3. MR 3488739. S2CID 119247836.
26. ^ Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2018). "Long gaps between primes". J. Amer. Math. Soc. 31 (1): 65–105. arXiv:1412.5029. doi:10.1090/jams/876. MR 3718451. S2CID 14487001.
27. ^
28. ^ Ford, Kevin; Maynard, James; Tao, Terence (2015-10-13). "Chains of large gaps between primes". arXiv:1511.04468 [math.NT].
29. ^ Cramér, Harald (1936). "On the order of magnitude of the difference between consecutive prime numbers" (PDF). Acta Arithmetica. 2: 23–46. doi:10.4064/aa-2-1-23-46. Archived from the original (PDF) on 2018-07-23. Retrieved 2016-06-27.
30. ^ A. E. Ingham, On the difference between consecutive primes, Quart. J. Math. (Oxford) 8, pp. 255-266 (1937).
31. ^ Sinha, Nilotpal Kanti (2010). "On a new property of primes that leads to a generalization of Cramer's conjecture". arXiv:1010.1399 [math.NT]..
32. ^ Granville, Andrew (1995). "Harald Cramér and the distribution of prime numbers" (PDF). Scandinavian Actuarial Journal. 1: 12–28. CiteSeerX 10.1.1.129.6847. doi:10.1080/03461238.1995.10413946..
33. ^ Granville, Andrew (1995). "Unexpected irregularities in the distribution of prime numbers" (PDF). Proceedings of the International Congress of Mathematicians. 1: 388–399. doi:10.1007/978-3-0348-9078-6_32. ISBN 978-3-0348-9897-3..
34. ^ Pintz, János (September 2007). "Cramér vs. Cramér: On Cramér's probabilistic model for primes". Functiones et Approximatio Commentarii Mathematici. 37 (2): 232–471. doi:10.7169/facm/1229619660.
35. ^ a b Guy (2004) §A8