Prime gap facts for kids

Prime gap frequency distribution for primes up to 1.6 billion. Peaks occur at multiples of 6.

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)-st 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
Numerical results
Further results
- Upper bounds
- Lower bounds
Conjectures about gaps between primes
As an arithmetic function
See also

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

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 1307674368000.

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 of ${\textstyle \frac{g_n}{\ln(p_n)}}$ 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 16,045,848, with 385,713-digit probable primes and merit M = 18.067, found by Andreas Höglund in March 2024. The largest known prime gap with identified proven primes as gap ends has length 1,113,106 and merit 25.90, with 18,662-digit primes found by P. Cami, M. Jansen and J. K. Andersen.

As of September 2022^[update], 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 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227. The prime gap between it and the next prime is 8350.

Largest known merit values (as of October 2020^[update])
Merit	g_n	digits	p_n	Date	Discoverer
41.938784	08350	0087	see above	2017	Gapcoin
39.620154	15900	0175	3483347771 × 409#/0030 − 7016	2017	Dana Jacobsen
38.066960	18306	0209	0650094367 × 491#/2310 − 8936	2017	Dana Jacobsen
38.047893	35308	0404	0100054841 × 953#/0210 − 9670	2020	Seth Troisi
37.824126	08382	0097	0512950801 × 229#/5610 − 4138	2018	Dana Jacobsen

The Cramér–Shanks–Granville ratio is the ratio of g_n / (ln(p_n))². If we discard anomalously high values of the ratio for the primes 2, 3, 7, then the greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. Other record terms can be found at OEIS: A111943.

We say that g_n is a maximal gap, if g_m < g_n for all m < n. As of May 2024^[update], the largest known maximal prime gap has length 1572, found by Craig Loizides. It is the 82nd maximal prime gap, and it occurs after the prime 18571673432051830099. Other record (maximal) gap sizes can be found in OEIS: A005250, with the corresponding primes p_n in OEIS: A002386, and the values of n in OEIS: A005669. The sequence of maximal gaps up to the nth prime is conjectured to have about $2\ln n$ terms (see table below).

The 82 known maximal prime gaps

Number 1 to 27
#	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
26	222	122,164,747	6,957,876
27	234	189,695,659	10,539,432

Number 28 to 54
#	g_n	p_n	n
28	248	191,912,783	10,655,462
29	250	387,096,133	20,684,332
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

Number 55 to 82
#	g_n	p_n	n
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	1,175,661,926,421,598
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

Further results

Upper bounds

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

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 an upper bound on the length of prime gaps:

For every $\epsilon > 0$ , there is a number $N$ such that for all $n > N$

g_n < p_n\epsilon

One can also 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 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, and to θ = 3/4 + ε, for any ε > 0, by Chudakov.

A major improvement is due to Ingham, who showed that for some positive constant c,

\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. 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).

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

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 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. 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 show that for any m there exists a bounded interval with an infinite number of translations each of which containing m prime numbers. 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.

Lower bounds

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

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

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 of n, improving the results of 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.

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.

Lower bounds for chains of primes have also been determined.

Conjectures about gaps between primes

Prime gap function

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. (In fact this result needs only the weaker Lindelöf hypothesis, if one can tolerate an infinitesimally larger exponent.) 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 \ge 1.

If this conjecture is true, then the function $g_n = p_{n+1} - p_n$ satisfies $g_n < (\log p_n)^2 - \log p_n \text{ for all } n > 4.$ It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz 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, under Oppermann's conjecture there exists $m$ (probably $m=30$ ) for which every natural number $n > m$ satisfies $g_n < \sqrt{p_n}.$

Andrica's conjecture, which is a weaker conjecture than Oppermann's, states that

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

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. The function is neither multiplicative nor additive.