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 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

As of 2016 the largest known prime gap with identified probable prime gap ends has length 4680156, with 99750-digit probable primes found by Martin Raab. This gap has merit M=20.3767 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.^[2]^[3]

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.^[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 .

**Largest known merit values** (As of November 2016)^[6]^[7]^[8]
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.^[6] The endpoints are 127-digit primes.

The Cramér–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
#	g_n	p_n
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
#	g_n	p_n
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
#	g_n	p_n
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^[9] 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,^[10] and to θ = 3/4 + ε, for any ε > 0, by Chudakov.^[11]

A major improvement is due to Ingham,^[12] 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.^[13] 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).^[14]

A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.^[15]

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^[16] 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.^[17] A Polymath Project collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013.^[18] 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.^[19] Using Maynard's ideas, the Polymath project improved the bound to 246;^[18]^[20] assuming the Elliott–Halberstam conjecture and its generalized form, N has been reduced to 12 and 6, respectively.^[18]

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.^[21]

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.^[22] This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.^[23]^[24]

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.^[25]

Lower bounds for chains of primes have also been determined.^[26]

Conjectures about gaps between primes

Prime gap function

Even better results are possible under the Riemann hypothesis. Harald Cramér proved^[27] 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 $g_{n}<(\log p_{n})^{2}-\log p_{n}{\text{ for all }}n>4.$ ^[28] It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz^[29]^[30]^[31] 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^[32]

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.^[32] The function is neither multiplicative nor additive.

References

↑ "Hidden structure in the randomness of the prime number sequence?", S. Ares & M. Castro, 2005
↑ 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. 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. doi:10.1216/rmj-2010-40-1-117. Zbl 1201.11111.
↑ Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones Mathematicae. 15 (2): 164–170. 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. doi:10.4007/annals.2014.179.3.7. MR 3171761.
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. doi:10.4007/annals.2015.181.1.7. MR 3272929.
↑ 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.
↑ 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. arXiv:1408.4505. doi:10.4007/annals.2016.183.3.4. MR 3488740.
↑ 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.
↑ 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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