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Prime gap

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.

$ 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 OEIS).

By the definition of gn the following sum can be stated 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) $In the opposite direction, the twin prime conjecture asserts that gn = 2 for infinitely many integers n.

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 which 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 2014 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 gn is a maximal gap if gm < gn 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 OEIS A002386.

Usually the ratio of gn / ln(pn) is called the merit of the gap gn . 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 OEIS A111870.

The Cramer-Shanks-Granville ratio is the ratio of $ gn / (ln(pn))^2 $ .[6] The greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. Other record terms can be found at OEIS 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 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 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 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 showed that one may choose θ = 7/12.^[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 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.[18][16] 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

Robert Rankin, improving results by Erik Westzynthius and Paul Erdős, 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: he showed that one can take any constant c < eγ, where γ is the Euler–Mascheroni constant. The value of the constant c was later improved to any constant c < 2eγ.[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 independently by Ford-Green-Konyagin-Tao and James Maynard, in the positive, by two papers respectively sent to arXiv in 2014.[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]
Conjectures about gaps between primes

Even better results are possible if it is assumed that the Riemann hypothesis is true. Harald Cramér proved that, under this assumption, the gap gn satisfies

$ g_n = O(\sqrt{p_n} \ln 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((\ln p_n)^2\right). $

At the moment, the numerical evidence seems to point in this direction. See Cramér's conjecture for more details.

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 $. [24]

This is one of the strongest upper bound ever conjectured for prime gaps. Moreover, this conjecture implies Cramér's conjecture in a strong form and would be consistent with Daniel Shanks conjectured asymptotic equality of record gaps.[25]

By using tables of maximal gaps, Firoozbakht's conjecture has been verified for all primes below 4×10¹⁸.[26]

Mean while, the Oppermann's conjecture is a conjecture which 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 to 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
Portal icon Mathematics portal

Bonse's inequality
Gaussian moat
Twin prime

References

Andersen, Jens Kruse. "The Top-20 Prime Gaps". Retrieved 2014-06-13.
Largest known prime gap
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.
NEW PRIME GAP OF MAXIMUM KNOWN MERIT
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.
"Primes in Tuples II". ArXiv. Retrieved 2013-11-23.
Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761.
"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.
DHJ Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences 1 (12). 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.
Guy (2004) §A8
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Kourbatov, Alexei. "prime Gaps: Firoozbakht Conjecture".

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

Feliksiak, Jan (2013). The Symphony Of Primes, Distribution Of Primes And Riemann's Hypothesis. Xlibris. ISBN 978-1479765584.