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.,

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

For example, since the first few primes are 2, 3, 5, 7, 11..., we have $g_1=1$, $g_2=g_3=2$, $g_4=4$. 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, ... Template: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 $g_2$ and $g_3$ 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

$n!+2,n!+3,\dots ,n!+n,$

the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of Template:Nowrap 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 $g_m\ge N$.

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 1 307 674 368 000.

The average gap between primes increases as the natural logarithm of these primes, and therefore the ratio of the prime gap to the primes 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 $e^{-k}$; 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.

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

Numerical results

Usually the ratio $g_n/(\ln p_n)$ is called the merit of the gap $g_n$. Informally, the merit of a gap $g_n$ can be thought of as the ratio of the size of the gap compared to the average prime gap sizes in the vicinity of $p_n$.

The largest known prime gap with identified probable prime gap ends has length Template:Val, with Template:Val-digit probable primes and merit Template:Nowrap, found by Andreas Höglund in Template:Date. The largest known prime gap with identified proven primes as gap ends has length Template:Val and merit 25.90, with Template:Val-digit primes found by P. Cami, M. Jansen and J. K. Andersen.

Template:As of, 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 Template:Zwsp. The prime gap between it and the next prime is 8350.

Largest known merit values (Template:As of)

Merit	gn	digits	pn	Date	Discoverer
Template:Val	8350	Template:087	Template:0see above	2017	Gapcoin
Template:Val	1854	Template:021	Template:0see below	2026	Robert Smith
Template:Val	15900	Template:0175	Template:0Template:Val × 409#/ 30 − 7016	2017	Dana Jacobsen
Template:Val	14496	Template:0164	Template:0Template:Val * 373# / 210 - 6502	2021	Craig Loizides
Template:Val	58044	Template:0656	Template:0Template:Val * 1549# / 46410 - 32632	2024	Robert Smith

The Cramér–Shanks–Granville ratio is the ratio $g_n/(\ln p_n{)}^2$. If we discard anomalously high values of the ratio for the primes 2, 3, and 7, then the greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. For comparison, the gap discovered by the Gapcoin network (with merit 41.9) has a Cramér–Shanks–Granville ratio of just 0.2058791. Other record terms can be found at Template:OEIS2C.

We say that $g_n$ is a maximal gap, if $g_m<g_n$ for all $m<n$. Template:As of, the largest known maximal prime gap has length 1854, found by Robert Smith, using code by Brian Kehrig. It is the 85th maximal prime gap, and it occurs after the prime 101412319996363309069. Other record (maximal) gap sizes can be found in Template:OEIS2C, with the corresponding primes $p_n$ in Template:OEIS2C, and the values of $n$ in Template:OEIS2C. The sequence of maximal gaps up to the $n$-th prime is conjectured to have about $2\ln n$ terms.

Template:Col-begin |+ The 85 known maximal prime gaps |-style="vertical-align: top;" |

Gaps 1 to 29

#	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
28	248	191,912,783	10,655,462
29	250	387,096,133	20,684,332

Gaps 30 to 58

#	gn	pn	n
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
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

Gaps 59 to 85

#	gn	pn	n
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
81	1,552	18,470,057,946,260,698,231	426,181,820,436,140,029
82	1,572	18,571,673,432,051,830,099	428,472,240,920,394,477
83	1,676	20,733,746,510,561,442,863	477,141,032,543,986,017
84	1,724	68,068,810,283,234,182,907	1,524,717,378,371,224,128
85	1,854	101,412,319,996,363,309,069	2,251,483,061,895,611,799

Further results

Upper bounds

Bertrand's postulate, proven in 1852, states that there is always a prime number between $k$ and $2k$ (for all k ≥ 2), so in particular $p_{n+1}<2p_n$, which means $g_n<p_n$.

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 $\ln p$, the natural logarithm of $p$, 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 that the gaps get arbitrarily smaller in proportion to the primes: the quotient

$\underset{n\to \mathrm{\infty }}{lim}\frac{g_n}{p_n}=0.$

In other words (by definition of a limit), for every $\epsilon >0$, there is a number $N$ such that for all $n>N$,

$g_n<p_n\epsilon$

Hoheisel (1930) was the first to show a sublinear dependence; that there exists a constant $\theta <1$ such that

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

hence showing that for sufficiently large $n$,

$g_n<{p_n}^{\theta }$

Hoheisel obtained the possible value 32999/33000 for $\theta$. This was improved to 249/250 by Heilbronn, and to $\theta =3/4+\epsilon$, for any $\epsilon >0$, by Chudakov.

A major improvement is due to Ingham, 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 Template:Nowrap is admissible, one obtains that θ may be any number greater than 5/8.

Since 5/8+ε < 2/3, and the gap between consecutive cubes is of the order of $n^{2/3}$, it follows that there is always a prime number between n³ and Template:Nowrap, if n is sufficiently large. In 2016, Dudek gave an explicit version of Ingham's result: there are primes between consecutive cubes for all $n>e^{e^{33.217}}\approx 1.01\cdot 10^{115809466034000}$.

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 Template:Nowrap 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 Template:Nowrap.

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

The above describes limits on all gaps; another area of interest is the minimum gap size. The twin prime conjecture asserts that there are always more gaps of size 2, but remains unproven. In 2005, Daniel Goldston, János Pintz and Cem Yıldırım proved that

$\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{g_n}{\log p_n}=0$

and 2 years later improved this to

$\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{g_n}{\sqrt{\log p_n}(\log \log p_n{)}^2}<\mathrm{\infty }.$

In 2013, Yitang Zhang proved that

$\underset{n\to \mathrm{\infty }}{lim\; inf}{g}_n<7\cdot 10^7,$

meaning that there are infinitely many gaps that do not exceed 70 million. A Polymath Project collaborative effort to optimize Zhang's bound managed to lower the bound to 4680 on July 20, 2013. In November 2013, James Maynard introduced a new refinement of the GPY sieve, allowing him to reduce the bound to 600 and also show that the gaps between primes m apart are bounded for all m. That is, for any m there exists a bound Δ_m such that Template:Nowrap for infinitely many n. Using Maynard's ideas, the Polymath project improved the bound to 246; assuming the Elliott–Halberstam conjecture and its generalized form, the bound has been reduced to 12 and 6, respectively. The current best-known bound is 240.

Lower bounds

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

$\underset{n\to \mathrm{\infty }}{lim\; sup}\frac{g_n}{\log p_n}=\mathrm{\infty }.$

In 1938, Robert Rankin proved the existence of a constant Template:Nowrap 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 of n, improving the results of Westzynthius and Paul Erdős. He later showed that one can take any constant Template:Nowrap, where γ is the Euler–Mascheroni constant. The value of the constant c was improved in 1997 to any value less than 2e^γ.

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. This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.

The result was further improved to

$g_n>\frac{c\log n\log \log n\log \log \log \log n}{\log \log \log n}$

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

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

In 2026, a proof and formalization in Lean claiming authorship by AI model GPT-5.6 Sol was uploaded to Github, positing an improvement on the bound to:

$g_n>\frac{c\log n\log \log n}{\log \log \log \log n}$

The claimed proof has not yet been fully verified.

Lower bounds for chains of primes have also been determined.

Conjectures about gaps between primes

As described above, the best proven bound on gap sizes is Template:Nowrap (for n sufficiently large; we do not worry about Template:Nowrap or Template:Nowrap), but it is observed that even maximal gaps are significantly smaller than that, leading to a plethora of unproven conjectures.

The first group hypothesize that the exponent can be reduced to Template:Nowrap.

Both Legendre's conjecture that there always exists a prime between consecutive perfect squares and Andrica's conjecture that the difference of square roots of consecutive primes is bounded by 1 imply that

$g_n<2\sqrt{p_n}.$

Oppermann's conjecture makes the stronger claim that, for sufficiently large n (probably Template:Nowrap),

$g_n<\sqrt{p_n}.$

All of these remain unproved. Harald Cramér came close, proving that the Riemann hypothesis implies the gap g_n satisfies

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

using the big O notation. (In fact this result needs only the weaker Lindelöf hypothesis, if one can tolerate an exponent infinitesimally larger than 1/2.)

Dudek also proved an explicit version of Cramer's result (also assuming Riemann hypothesis) that is, for all n ≥ 2,

$g_n<\frac{4}{\pi }\sqrt{p_n}\log p_n$

Prime gap function

As noted by Dudek, for sufficiently large n one may reduce the constant $4/\pi$ to be any constant greater than 1.

In the same article, Cramér conjectured that the gaps are far smaller. Roughly speaking, Cramér's conjecture states that

$g_n=O((\log p_n{)}^2),$

a polylogarithmic growth rate slower than any exponent Template:Nowrap.

Cramér's model, under which he made the conjecture, was oversimplified (assuming some events are statistically independent when they are dependent) and thus not very accurate (see Cramér's conjecture), but after further investigations, new heuristics were found which became strong evidence that the conjecture is true.

As this matches the observed growth rate of prime gaps, there are a number of similar conjectures. Firoozbakht's conjecture is slightly stronger, stating that p_n^1/n is a strictly decreasing function of n, i.e., for all n ≥ 1,

$(p_{n+1}{)}^{1/(n+1)}<(p_n{)}^{1/n}$

If this conjecture is true, then Template:Nowrap. It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz, which suggest that Template:Nowrap infinitely often for any Template:Nowrap, where γ denotes the Euler–Mascheroni constant.

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 the improvements on Zhang's result discussed above prove that it is true for at least one (currently unknown) value of k ≤ 246.

In her 2026 paper, Julia Stadlmann of the University of Illinois at Urbana-Champaign improved the bound to 240, using the Bombieri–Vinogradov theorem with newer equidistribution estimates for smooth moduli, with Thomas Bloom noting the result can be improved a little further with more effort.

GPT 5.6 also found a proof to improve the bound to 240 . This was earlier than Julia Stadlmann's paper.

As an arithmetic function

The gap g_n between the nth and (n + 1)th prime numbers is an example of an arithmetic function. In this context it is usually denoted d_n and called the prime difference function. The function is neither multiplicative nor additive. Template:Clear

See also

• Bonse's inequality
• Gaussian moat
• Twin prime

References

• Template:Cite book

Further reading

• Template:Cite journal
• Template:Cite journal

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.
• Template:MathWorld
• Template:Planetmath reference
• 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
• 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.
• Birke Heeren, [1] Here you find a paper on how to calculate large gaps.