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 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 the 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 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

n!+2,n!+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 − 1. It follows that there are gaps between primes that are arbitrarily large, that is, for any integer N, there is an integer m with $g m \geq N$ .

In reality, 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.

Although the average gap between primes increases as the natural logarithm of the integer, the ratio of the prime gap to the integers involved decreases (and is asymptotically zero). This is a consequence of the prime number theorem. On the other hand, 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.

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

Numerical results

Usually the ratio of g_n / ln(p_n) is called the merit of the gap g_n .

As of March 2017 the largest known prime gap with identified probable prime gap ends has length 5103138, with 216849-digit probable primes found by Robert W. Smith.^[2] This gap has merit M=10.2203. 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.^[3]^[4]

We say that g_n is a maximal gap, if g_m < g_n for all m < n. As of August 2016 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.^[5] Other record maximal gap terms can be found at A002386.

In 1931, E. Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,^[6]

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

**Largest known merit values** (As of November 2016)^[7]^[8]^[9]
Merit	g_n	digits	p_n	Date	Discoverer
36.858288	10716	127	7910896513*283#/30 - 6480	2016	Dana Jacobsen
36.590183	13692	163	1037600971*383#/210 - 8776	2016	Dana Jacobsen
36.420568	26892	321	59740589*757#/210 - 14302	2016	Dana Jacobsen
35.424459	66520	816	1931*1933#/7230 - 30244	2012	Michiel Jansen
35.310308	1476	19	1425172824437699411	2009	Tomás Oliveira e Silva

As of November 2016, the largest known merit value, as discovered by D. Jacobsen, is 10716 / ln(7910896513*283#/30 - 6480) ≈ 36.858288 where 283# indicates the primorial of 283.^[7] The endpoints are 127-digit primes.

The Cramér–Shanks–Granville ratio is the ratio of g_n / (ln(p_n))^2.^[7] 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

Number 1 to 25
#	g_n	p_n	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

Number 26 to 50
#	g_n	p_n	n
26	222	122,164,747	6,957,876
27	234	189,695,659	10,539,432
28	248	191,912,783	10,655,462
29	250	387,096,133	20,684,332
30	282	436,273,009	32,162,398
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

Number 51 to 75
#	g_n	p_n	n
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
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	1,175,662,926,421,598
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

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^[10] that there exists a constant θ < 1 such that

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

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,^[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

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

then

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

for any

\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

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

and 2 years later improved this^[17] 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.^[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 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 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.^[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

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.^[26]

Lower bounds for chains of primes have also been determined.^[27]

Conjectures about gaps between primes

Prime gap function

Even better results are possible under the Riemann hypothesis. Harald Cramér proved^[28] 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, Cramér's conjecture states 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\geq 1.

If this conjecture is true, then the function $g_{n}=p_{{n+1}}-p_{n}$ satisfies $g_{n}<(\log p_{n})^{2}-\log p_{n}{\text{ for all }}n>4.$ ^[29] It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz^[30]^[31]^[32] 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 on the order of

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

As a result, there is (under Oppermann's conjecture) m>0 (probably m=30) for which every natural n>m satisfies $g_{n}<{\sqrt {p_{n}}}.$

Andrica's conjecture, which is a weaker conjecture than Oppermann's, states that^[33]

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 result proves that it is true for at least one (currently unknown) value of k which is smaller than 70,000,000.

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.^[33] The function is neither multiplicative nor additive.

References

↑ "Hidden structure in the randomness of the prime number sequence?", S. Ares & M. Castro, 2005
↑ http://trnicely.net/index.html#TPG
↑ Andersen, Jens Kruse. "The Top-20 Prime Gaps". Retrieved 2014-06-13.
↑ A proven prime gap of 1113106
↑ Maximal Prime Gaps
↑ 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 .
1 2 3 NEW PRIME GAP OF MAXIMUM KNOWN MERIT
↑ Dynamic prime gap statistics
↑ TABLES OF PRIME GAPS
↑ Hoheisel, G. (1930). "Primzahlprobleme in der Analysis". Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. 33: 3–11. JFM 56.0172.02.
↑ Heilbronn, H. A. (1933). "Über den Primzahlsatz von Herrn Hoheisel". Mathematische Zeitschrift. 36 (1): 394–423. doi:10.1007/BF01188631.
↑ Tchudakoff, N. G. (1936). "On the difference between two neighboring prime numbers". Math. Sb. 1: 799–814.
↑ 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.
↑ Cheng, Yuan-You Fu-Rui (2010). "Explicit estimate on primes between consecutive cubes". Rocky Mt. J. Math. 40: 117–153. Zbl 1201.11111. doi:10.1216/rmj-2010-40-1-117.
↑ Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones Mathematicae. 15 (2): 164–170. Bibcode:1971InMat..15..164H. doi:10.1007/BF01418933.
↑ 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.
↑ Goldston, D. A.; Pintz, J.; Yildirim, C. Y. (2007). "Primes in Tuples II". arXiv:0710.2728  [math.NT].
↑ Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. MR 3171761. doi:10.4007/annals.2014.179.3.7.
1 2 3 "Bounded gaps between primes". Polymath. Retrieved 2013-07-21.
↑ Maynard, James (2015). "Small gaps between primes". Annals of Mathematics. 181 (1): 383–413. MR 3272929. doi:10.4007/annals.2015.181.1.7.
↑ D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences. 1 (12). MR 3373710. arXiv:1407.4897 . doi:10.1186/s40687-014-0012-7.
↑ Pintz, J. (1997). "Very large gaps between consecutive primes". J. Number Theory. 63 (2): 286–301. doi:10.1006/jnth.1997.2081.
↑ Erdős, Some of my favourite unsolved problems
↑ Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence (2016). "Large gaps between consecutive prime numbers". Ann. of Math. 183 (3): 935–974. MR 3488740. arXiv:1408.4505 . doi:10.4007/annals.2016.183.3.4.
↑ Maynard, James (2016). "Large gaps between primes". Ann. of Math. 183 (3): 915–933. MR 3488739. arXiv:1408.5110 . doi:10.4007/annals.2016.183.3.3.
↑ Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2015). "Long gaps between primes". arXiv:1412.5029  [math.NT].
↑ Ford, Kevin; Maynard, James; Tao, Terence (2015-10-13). "Chains of large gaps between primes". arXiv:1511.04468  [math.NT].
↑ Cramér, Harald (1936). "On the order of magnitude of the difference between consecutive prime numbers" (PDF). Acta Arithmetica. 2: 23–46.
↑ Sinha, Nilotpal Kanti (2010). "On a new property of primes that leads to a generalization of Cramer's conjecture". arXiv:1010.1399  [math.NT]. .
↑ Granville, A. (1995). "Harald Cramér and the distribution of prime numbers" (PDF). Scandinavian Actuarial Journal. 1: 12–28. .
↑ Granville, Andrew (1995). "Unexpected irregularities in the distribution of prime numbers" (PDF). Proceedings of the International Congress of Mathematicians. 1: 388–399. .
↑ Pintz, János (2007). "Cramér vs. Cramér: On Cramér's probabilistic model for primes". Funct. Approx. Comment. Math. 37 (2): 232–471.
1 2 Guy (2004) §A8

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

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". PlanetMath.
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.

Prime number classes
By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin
Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)
By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Almost prime Semiprime Interprime
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229
List of prime numbers

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