Prime gap

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

${\displaystyle n!+2,\;n!+3,\;\ldots ,\;n!+n}$ 

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 ≥ 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.^[2]

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

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.^[3] 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.^[4][5]

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 2​9​3​7​0​3​2​3​4​0​6​8​0​2​2​5​9​0​1​5​8​7​2​3​7​6​6​1​0​4​4​1​9​4​6​3​4​2​5​7​0​9​0​7​5​5​7​4​8​1​1​7​6​2​0​9​8​5​8​8​7​9​8​2​1​7​8​9​5​7​2​8​8​5​8​6​7​6​7​2​8​1​4​3​2​2​7. The prime gap between it and the next prime is 8350.^[6][7]

The Cramér–Shanks–Granville ratio is the ratio of g_n / (ln(p_n))².^[6] 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 October 2024^[update], the largest known maximal prime gap has length 1676, found by Brian Kehrig. It is the 83rd maximal prime gap, and it occurs after the prime 20733746510561442863.^[11] 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 ${\displaystyle 2\ln n}$  terms^[12] (see table below).

Bertrand's postulate, proven 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, 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

${\displaystyle \lim _{n\to \infty }{\frac {g_{n}}{p_{n}}}=0.}$ 

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

${\displaystyle g_{n}<p_{n}\epsilon }$ .

Hoheisel (1930) was the first to show^[13] a sublinear dependence; that there exists a constant θ < 1 such that

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

hence showing that

${\displaystyle g_{n}<p_{n}^{\theta },}$ 

for sufficiently large n.

Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,^[14] and to θ = 3/4 + ε, for any ε > 0, by Chudakov.^[15]

A major improvement is due to Ingham,^[16] who showed that for some positive constant c,

if ${\displaystyle \zeta (1/2+it)=O(t^{c})}$  then ${\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log(x)}}}$  for any ${\displaystyle \theta >(1+4c)/(2+4c).}$ 

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.^[17] 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).^[18]

A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.^[19]

The above describes limits on all gaps; another are 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

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0}$ 

and 2 years later improved this^[20] to

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty .}$ 

In 2013, Yitang Zhang proved that

${\displaystyle \liminf _{n\to \infty }g_{n}<7\cdot 10^{7},}$ 

meaning that there are infinitely many gaps that do not exceed 70 million.^[21] A Polymath Project collaborative effort to optimize Zhang's bound managed to lower the bound to 4680 on July 20, 2013.^[22] 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^{[incomprehensible]}.^[23] Using Maynard's ideas, the Polymath project improved the bound to 246;^[22][24] assuming the Elliott–Halberstam conjecture and its generalized form, the bound has been reduced to 12 and 6, respectively.^[22]

In 1931, Erik Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,^[2]

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

In 1938, Robert Rankin proved the existence of a constant c > 0 such that the inequality

${\displaystyle g_{n}>{\frac {c\ \log n\ \log \log n\ \log \log \log \log n}{(\log \log \log n)^{2}}}}$ 

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.^[25]

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.^[26] This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.^[27][28]

The result was further improved to

${\displaystyle g_{n}>{\frac {c\ \log n\ \log \log n\ \log \log \log \log n}{\log \log \log n}}}$ 

for infinitely many values of n by Ford–Green–Konyagin–Maynard–Tao.^[29]

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.^[30]

Lower bounds for chains of primes have also been determined.^[31]

As described above, the best proven bound on gap sizes is ${\displaystyle g_{n}<p_{n}^{0.525}}$  (for ${\displaystyle n}$  sufficiently large; we do not worry about ${\displaystyle 5-3>3^{0.525}}$  or ${\displaystyle 29-23>23^{0.525}}$ ), 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 ${\displaystyle \theta =0.5}$ .

Legendre's conjecture that there always exists a prime between successive square numbers implies that ${\displaystyle g_{n}=O({\sqrt {p_{n}}})}$ . Andrica's conjecture states that^[32]

${\displaystyle g_{n}<2{\sqrt {p_{n}}}+1.}$ 

Oppermann's conjecture makes the stronger claim that, for sufficiently large ${\displaystyle n}$  (probably ${\displaystyle n>30}$ ),

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

All of these remain unproved. Harald Cramér came close, proving^[33] that the Riemann hypothesis implies the gap g_n satisfies

${\displaystyle g_{n}=O({\sqrt {p_{n}}}\log p_{n}),}$ 

using the big O notation. (In fact this result needs only the weaker Lindelöf hypothesis, if one can tolerate an infinitesimally larger exponent.^[34])

 

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

${\displaystyle g_{n}=O\!\left((\log p_{n})^{2}\right)\!,}$ 

a polylogarithmic growth rate slower than any exponent ${\displaystyle \theta >0}$ .

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 ${\displaystyle p_{n}^{1/n}\!}$  is a strictly decreasing function of n, i.e.,

${\displaystyle p_{n+1}^{1/(n+1)}\!<p_{n}^{1/n}{\text{ for all }}n\geq 1.}$ 

If this conjecture were true, then ${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}-1{\text{ for all }}n>9.}$ ^[35][36] It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz^[37][38][39] which suggest that ${\displaystyle g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}>(1.1229-\varepsilon )(\log p_{n})^{2}}$  infinitely often for any ${\displaystyle \varepsilon >0,}$  where ${\displaystyle \gamma }$  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.

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.

• Bonse's inequality
• Gaussian moat
• Twin prime

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

• Soundararajan, Kannan (2007). "Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım". Bull. Am. Math. Soc. New Series. 44 (1): 1–18. arXiv:math/0605696. doi:10.1090/s0273-0979-06-01142-6. S2CID 119611838. Zbl 1193.11086.
• Mihăilescu, Preda (June 2014). "On some conjectures in additive number theory" (PDF). EMS Newsletter (92): 13–16. doi:10.4171/NEWS. hdl:2117/17085. ISSN 1027-488X.

• 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
• 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 the large prime number gaps and a paper on how to calculate such large gaps.