Prime gap

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

$g_n = p_{n + 1} - p_n.\$

We have g₁ = 1, g₂ = g₃ = 2, and g₄ = 4. The sequence (g_n) 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 OEIS).

By the definition of g_n every prime can be written as

$p_{n+1} = 2 + \sum_{i=1}^n g_i$ .

Simple observations

The first, smallest, and only odd prime gap is 1 between the only even prime number, 2, and the first odd prime, 3. All other prime gaps are even. There is only one pair of gaps between three consecutive odd natural numbers for which all are prime. These gaps are g₂ and g₃ between the primes 3, 5, and 7.

For any prime number P, we write P# for P primorial, that is, the product of all prime numbers up to and including P. If Q is the prime number following P, then the sequence

$P\#+2, P\#+3,\ldots,P\#+(Q-1)$

is a sequence of Q − 2 consecutive composite integers, so here there is a prime gap of at least length Q − 1. Therefore, there exist gaps between primes that are arbitrarily large, i.e., for any prime number P, there is an integer n with g_n ≥ P. (This is seen by choosing n so that p_n is the greatest prime number less than P# + 2.) Another way to see that arbitrarily large prime gaps must exist is the fact that the density of primes approaches zero, according to the prime number theorem. In fact, by this theorem, P# is very roughly a number the size of exp(P), and near exp(P) the average distance between consecutive primes is P.

In reality, prime gaps of P numbers can occur at numbers much smaller than P#. For instance, the smallest sequence of 71 consecutive composite numbers occurs between 31398 and 31468, whereas 71# has twenty-seven digits – its full decimal expansion being 557940830126698960967415390.

Although the average gap between primes increases as the natural logarithm of the integer, the ratio of the maximum prime gap to the integers involved also increases as larger and larger numbers and gaps are encountered.

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

Numerical results

Prime gap function

As of 2016 the largest known prime gap with identified probable prime gap ends has length 3311852, with 97953-digit probable primes found by M. Jansen and J. K. Andersen.^[1][2] The largest known prime gap with identified proven primes as gap ends has length 1113106, with 18662-digit primes found by P. Cami, M. Jansen and J. K. Andersen.^[1][3]

We say that g_n is a maximal gap, if g_m < g_n for all m < n. As of June 2014 the largest known maximal gap has length 1476, found by Tomás Oliveira e Silva. It is the 75th maximal gap, and it occurs after the prime 1425172824437699411.^[4] Other record maximal gap terms can be found at  A002386.

Usually the ratio of g_n / ln(p_n) is called the merit of the gap g_n . In 1931, E. Westzynthius proved that prime gaps grow more than logarithmically. That is,^[5]

$\limsup_{n\to\infty}\frac{g_n}{\log p_n}=\infty.$

As of January 2012, the largest known merit value, as discovered by M. Jansen, is 66520 / ln(1931*1933#/7230 - 30244) ≈ 35.4244594 where 1933# indicates the primorial of 1933. This number, 1931*1933#/7230 - 30244, is an 816-digit prime. The next largest known merit value is 1476 / ln(1425172824437699411) ≈ 35.31.^[1][6] This prime with the gap of 1476 is also the 75th maximal gap (the last one in the table below). Other record merit terms can be found at  A111870.

The Cramer-Shanks-Granville ratio is the ratio of g_n / (ln(p_n))^2.^[6] The greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. Other record terms can be found at  A111943.

The first 75 maximal gaps (n is not listed)

Number 1 to 25 # gn pn 1 1 2 2 2 3 3 4 7 4 6 23 5 8 89 6 14 113 7 18 523 8 20 887 9 22 1,129 10 34 1,327 11 36 9,551 12 44 15,683 13 52 19,609 14 72 31,397 15 86 155,921 16 96 360,653 17 112 370,261 18 114 492,113 19 118 1,349,533 20 132 1,357,201 21 148 2,010,733 22 154 4,652,353 23 180 17,051,707 24 210 20,831,323 25 220 47,326,693

Number 26 to 50 # gn pn 26 222 122,164,747 27 234 189,695,659 28 248 191,912,783 29 250 387,096,133 30 282 436,273,009 31 288 1,294,268,491 32 292 1,453,168,141 33 320 2,300,942,549 34 336 3,842,610,773 35 354 4,302,407,359 36 382 10,726,904,659 37 384 20,678,048,297 38 394 22,367,084,959 39 456 25,056,082,087 40 464 42,652,618,343 41 468 127,976,334,671 42 474 182,226,896,239 43 486 241,160,624,143 44 490 297,501,075,799 45 500 303,371,455,241 46 514 304,599,508,537 47 516 416,608,695,821 48 532 461,690,510,011 49 534 614,487,453,523 50 540 738,832,927,927

Number 51 to 75 # gn pn 51 582 1,346,294,310,749 52 588 1,408,695,493,609 53 602 1,968,188,556,461 54 652 2,614,941,710,599 55 674 7,177,162,611,713 56 716 13,829,048,559,701 57 766 19,581,334,192,423 58 778 42,842,283,925,351 59 804 90,874,329,411,493 60 806 171,231,342,420,521 61 906 218,209,405,436,543 62 916 1,189,459,969,825,483 63 924 1,686,994,940,955,803 64 1,132 1,693,182,318,746,371 65 1,184 43,841,547,845,541,059 66 1,198 55,350,776,431,903,243 67 1,220 80,873,624,627,234,849 68 1,224 203,986,478,517,455,989 69 1,248 218,034,721,194,214,273 70 1,272 305,405,826,521,087,869 71 1,328 352,521,223,451,364,323 72 1,356 401,429,925,999,153,707 73 1,370 418,032,645,936,712,127 74 1,442 804,212,830,686,677,669 75 1,476 1,425,172,824,437,699,411

Further results

Upper bounds

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

The prime number theorem proved in 1896 says that the "average length" of the gap between a prime p and the next prime is ln(p). The actual length of the gap might be much more or less than this. However, from the prime number theorem one can also deduce an upper bound on the length of prime gaps: for every ε > 0, there is a number N such that g_n < εp_n for all n > N.

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

$\lim_{n\to\infty}\frac{g_n}{p_n}=0$.

Hoheisel (1930) was the first to show^[7] that there exists a constant θ < 1 such that

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

hence showing that

$g_n<p_n^\theta,\,$

for sufficiently large n.

Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,^[8] and to θ = 3/4 + ε, for any ε > 0, by Chudakov.^[9]

A major improvement is due to Ingham,^[10] who showed that if

$\zeta(1/2 + it)=O(t^c)\,$

for some positive constant c, where O refers to the big O notation, then

$\pi(x + x^\theta) - \pi(x) \sim \frac{x^\theta}{\log(x)}$

for any θ > (1 + 4c)/(2 + 4c). Here, as usual, 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.^[11] 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).^[12]

A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.^[13]

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

$\liminf_{n\to\infty}\frac{g_n}{\log p_n}=0$

and 2 years later improved it^[14] to

$\liminf_{n\to\infty}\frac{g_n}{\sqrt{\log p_n}(\log\log p_n)^2}<\infty.$

In 2013, Yitang Zhang proved that

$\liminf_{n\to\infty} g_n < 7\cdot 10^7$,

meaning that there are infinitely many gaps that do not exceed 70 million.^[15] A Polymath Project collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013.^[16] 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 containing m prime numbers.^[17] Using Maynard's ideas, the Polymath project has since improved the bound to 246.^[16][18] Further, assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that N has been reduced to 12 and 6, respectively.^[16]

Lower bounds

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

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

holds for infinitely many values n, improving results by Erik Westzynthius and Paul Erdős. He later showed that one can take any constant c < e^γ, where γ is the Euler–Mascheroni constant. The value of the constant c was improved in 1997 to any value less than 2e^γ.^[19]

Paul Erdős offered a $5,000 prize for a proof or disproof that the constant c in the above inequality may be taken arbitrarily large.^[20] This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.^[21][22]

The result was further improved to

$g_n \gg \frac{\log n\log\log n\log\log\log\log n}{\log\log\log n}$

for infinitely many values of n by Ford–Green–Konyagin–Maynard–Tao.^[23]

Lower bounds for chains of primes have also been determined.^[24]

Conjectures about gaps between primes

Even better results are possible under the Riemann hypothesis. Harald Cramér proved that the Riemann hypothesis implies the gap g_n satisfies

$g_n = O(\sqrt{p_n} \log p_n),$

using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that

$g_n = O\left((\log p_n)^2\right).$

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

$p_{n+1}^{1/(n+1)} < p_n^{1/n} \text{ for all } n \ge 1.$

If this conjecture is true, then the function $g_n = p_{n+1} - p_n$ satisfies $g_n < (\log p_n)^2 - \log p_n \text{ for all } n > 4.$^[25] It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz^[26][27][28] which suggest that $g_n > \frac{2-\varepsilon}{e^\gamma}(\log p_n)^2$ infinitely often for any $\varepsilon>0,$ where $\gamma$ denotes the Euler–Mascheroni constant.

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

$g_n < \sqrt{p_n}\,$.

Andrica's conjecture, which is a weaker conjecture than Oppermann's, states that^[20]

$g_n < 2\sqrt{p_n} + 1.\,$

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

As an arithmetic function

The gap g_n between the nth and (n + 1)st prime numbers is an example of an arithmetic function. In this context it is usually denoted d_n and called the prime difference function.^[20] The function is neither multiplicative nor additive.

See also

• Bonse's inequality
• Gaussian moat
• Twin prime

References

1. Andersen, Jens Kruse. "The Top-20 Prime Gaps". Retrieved 2014-06-13.
2. Largest known prime gap
3. A proven prime gap of 1113106
4. Maximal Prime Gaps
5. 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.
6. NEW PRIME GAP OF MAXIMUM KNOWN MERIT
7. Hoheisel, G. (1930). "Primzahlprobleme in der Analysis". Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 33: 3–11. JFM 56.0172.02.
8. Heilbronn, H. A. (1933). "Über den Primzahlsatz von Herrn Hoheisel". Mathematische Zeitschrift 36 (1): 394–423. doi:10.1007/BF01188631.
9. Tchudakoff, N. G. (1936). "On the difference between two neighboring prime numbers". Math. Sb. 1: 799–814.
10. Ingham, A. E. (1937). "On the difference between consecutive primes". Quarterly Journal of Mathematics. Oxford Series 8 (1): 255–266. doi:10.1093/qmath/os-8.1.255.
11. Cheng, Yuan-You Fu-Rui (2010). "Explicit estimate on primes between consecutive cubes". Rocky Mt. J. Math. 40: 117–153. doi:10.1216/rmj-2010-40-1-117. Zbl 1201.11111.
12. Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones Mathematicae 15 (2): 164–170. doi:10.1007/BF01418933.
13. 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.
14. "Primes in Tuples II". ArXiv. Retrieved 2013-11-23.
15. Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761.
16. "Bounded gaps between primes". Polymath. Retrieved 2013-07-21.
17. Maynard, James (2015). "Small gaps between primes". Annals of Mathematics 181 (1): 383–413. doi:10.4007/annals.2015.181.1.7. MR 3272929.
18. 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.
19. Pintz, J. (1997). "Very large gaps between consecutive primes". J. Number Theory 63 (2): 286–301. doi:10.1006/jnth.1997.2081.
20. Guy (2004) §A8
21. 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.
22. Maynard, James (2016). "Large gaps between primes" 183 (3): 915–933. arXiv:1408.5110v1. doi:10.4007/annals.2016.183.3.3.
23. Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2015). "Long gaps between primes". arXiv:1412.5029.
24. Ford, Kevin; Manyard, James; Tao, Terence (2015-10-13). "Chains of large gaps between primes". arXiv:1511.04468.
25. Sinha, Nilotpal Kanti (2010), On a new property of primes that leads to a generalization of Cramer's conjecture, pp. 1–10, arXiv:1010.1399.
26. Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal 1: 12–28.
27. Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians 1: 388–399.
28. Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math. 37 (2): 232–471

• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 978-0-387-20860-2. Zbl 1058.11001.

Further reading

• Soundararajan, Kannan (2007). "Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım". Bull. Am. Math. Soc., New Ser. 44 (1): 1–18. doi:10.1090/s0273-0979-06-01142-6. Zbl 1193.11086.
• Mihăilescu, Preda (June 2014). "On some conjectures in additive number theory" (PDF). Newsletter of the European Mathematical Society (92): 13–16. doi:10.4171/NEWS. ISSN 1027-488X.

External links

• Thomas R. Nicely, Some Results of Computational Research in Prime Numbers -- Computational Number Theory. This reference web site includes a list of all first known occurrence prime gaps.
• Weisstein, Eric W., "Prime Difference Function", MathWorld.
• Prime Difference Function at PlanetMath.org.
• Armin Shams, Re-extending Chebyshev's theorem about Bertrand's conjecture, does not involve an 'arbitrarily big' constant as some other reported results.
• Chris Caldwell, Gaps Between Primes; an elementary introduction
• www.primegaps.com A study of the gaps between consecutive prime numbers
• Andrew Granville, Primes in Intervals of Bounded Length; overview of the results obtained so far up to and including James Maynard's work of November 2013.