Prime gap

A prime gap is the difference between two successive prime numbers. The n {\displaystyle n}-th prime gap, denoted g n {\displaystyle g_{n}} or g ( p n ) {\displaystyle g(p_{n})} is the difference between the ( n + 1 ) {\displaystyle (n+1)}th and the n {\displaystyle n}-th prime numbers, i.e.,

g n = p n + 1 − p n {\displaystyle 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 {\displaystyle g_{1}=1}, g 2 = g 3 = 2 {\displaystyle g_{2}=g_{3}=2}, g 4 = 4 {\displaystyle g_{4}=4}. The sequence g n {\displaystyle 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, ... (sequenceA001223in theOEIS).

By the definition of g n {\displaystyle g_{n}} every prime can be written as

p n + 1 = 2 + ∑ i = 1 n g i . {\displaystyle 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 {\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} between the primes 3, 5, and 7.

For any integer n {\displaystyle n}, the factorial n ! {\displaystyle n!} is the product of all positive integers up to and including n {\displaystyle n}. Then in the sequence

n ! + 2 , n ! + 3 , … , n ! + n , {\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 {\displaystyle n}. It follows that there are gaps between primes that are arbitrarily large, that is, for any integer N {\displaystyle N}, there is an integer m {\displaystyle m} with g m ≥ N {\displaystyle g_{m}\geq N}.

However, prime gaps of n {\displaystyle n} numbers can occur at numbers much smaller than n ! {\displaystyle 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 {\displaystyle k} to be e − k {\displaystyle 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 {\displaystyle g_{n}=2} for infinitely many integers n {\displaystyle n}.

Numerical results

Usually the ratio g n / ( ln ⁡ p n ) {\displaystyle g_{n}/(\ln p_{n})} is called the merit of the gap g n {\displaystyle g_{n}}. Informally, the merit of a gap g n {\displaystyle 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 {\displaystyle p_{n}}.

The largest known prime gap with identified probable prime gap ends has length 16045848, with 385713-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 1113106 and merit 25.90, with 18662-digit primes found by P. Cami, M. Jansen and J. K. Andersen.

As of May2026[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.

The Cramér–Shanks–Granville ratio is the ratio g n / ( ln ⁡ p n ) 2 {\displaystyle 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 OEIS:A111943.

We say that g n {\displaystyle g_{n}} is a maximal gap, if g m < g n {\displaystyle g_{m}<g_{n}} for all m < n {\displaystyle m<n}. As of May2026[update], 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 OEIS:A005250, with the corresponding primes p n {\displaystyle p_{n}} in OEIS:A002386, and the values of n {\displaystyle n} in OEIS:A005669. The sequence of maximal gaps up to the n {\displaystyle n}-th prime is conjectured to have about 2 ln ⁡ n {\displaystyle 2\ln n} terms.

Further results

Upper bounds

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

The prime number theorem, proven in 1896, says that the average length of the gap between a prime p {\displaystyle p} and the next prime will asymptotically approach ln ⁡ p {\displaystyle \ln p}, the natural logarithm of p {\displaystyle 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

lim n → ∞ g n p n = 0. {\displaystyle \lim _{n\to \infty }{\frac {g_{n}}{p_{n}}}=0.}

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

g n < p n ε {\displaystyle g_{n}<p_{n}\varepsilon }

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

π ( x + x θ ) − π ( x ) ∼ x θ log ⁡ x as x → ∞ , {\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log x}}{\text{ as }}x\to \infty ,}

hence showing that for sufficiently large n {\displaystyle n},

g n < p n θ {\displaystyle g_{n}<{p_{n}}^{\theta }}

Hoheisel obtained the possible value 32999/33000 for θ {\displaystyle \theta }. This was improved to 249/250 by Heilbronn, and to θ = 3 / 4 + ε {\displaystyle \theta =3/4+\varepsilon }, for any ε > 0 {\displaystyle \varepsilon >0}, by Chudakov.

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

if ζ ( 1 / 2 + i t ) = O ( t c ) {\displaystyle \zeta (1/2+it)=O(t^{c})} then π ( x + x θ ) − π ( x ) ∼ x θ log ⁡ x {\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log x}}} for any θ > ( 1 + 4 c ) / ( 2 + 4 c ) . {\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.

Since 5/8+ε < 2/3, and the gap between consecutive cubes is of the order of n 2 / 3 {\displaystyle n^{2/3}}, it follows that there is always a prime number between n3 and (n + 1)3, 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 ≈ 1.01 ⋅ 10 115809466034000 {\displaystyle 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 (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 = 0.583.

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

lim inf n → ∞ g n log ⁡ p n = 0 {\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0}

and 2 years later improved this to

lim inf n → ∞ g n log ⁡ p n ( log ⁡ log ⁡ p n ) 2 < ∞ . {\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty .}

In 2013, Yitang Zhang proved that

lim inf n → ∞ g n < 7 ⋅ 10 7 , {\displaystyle \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 also show that the gaps between primes m apart are bounded for all m. That is, for any m there exists a bound Δm such that pn+m − pn ≤ Δm 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.

Lower bounds

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

lim sup n → ∞ g n log ⁡ p n = ∞ . {\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

g n > c log ⁡ n log ⁡ log ⁡ n log ⁡ log ⁡ log ⁡ log ⁡ n ( log ⁡ log ⁡ log ⁡ n ) 2 {\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 < eγ, where γ is the Euler–Mascheroni constant. The value of the constant c was improved in 1997 to any value less than 2eγ.

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 > c log ⁡ n log ⁡ log ⁡ n log ⁡ log ⁡ log ⁡ log ⁡ n log ⁡ log ⁡ log ⁡ n {\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.

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

As described above, the best proven bound on gap sizes is gn < pn0.525 (for n sufficiently large; we do not worry about 5 − 3 > 30.525 or 29 − 23 > 230.525), 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 θ = 0.5.

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 p n . {\displaystyle g_{n}<2{\sqrt {p_{n}}}.}

Oppermann's conjecture makes the stronger claim that, for sufficiently large n (probably n ≥ 31),

g n < p n . {\displaystyle g_{n}<{\sqrt {p_{n}}}.}

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

g n = O ( p n log ⁡ p n ) , {\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 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 < 4 π p n log ⁡ p n {\displaystyle g_{n}<{\frac {4}{\pi }}{\sqrt {p_{n}}}\log p_{n}}

As noted by Dudek, for sufficiently large n one may reduce the constant 4 / π {\displaystyle 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 ) , {\displaystyle g_{n}=O\!\left((\log p_{n})^{2}\right)\!,}

a polylogarithmic growth rate slower than any exponent θ > 0.

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 pn1/n is a strictly decreasing function of n, i.e., for all n ≥ 1,

( p n + 1 ) 1 / ( n + 1 ) < ( p n ) 1 / n {\displaystyle (p_{n+1})^{1/(n+1)}<(p_{n})^{1/n}}

If this conjecture is true, then gn < (log pn)2 − log pn − 1 for all n ≥ 10. It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz, which suggest that gn > (2 − ϵ)e−γ(log pn)2 > (1.1229 − ϵ)(log pn)2 infinitely often for any ϵ > 0, 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 ofk, 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 gn between the nth and (n+1)th prime numbers is an example of an arithmetic function. In this context it is usually denoted dn and called the prime difference function. The function is neither multiplicative nor additive.

See also

• Bonse's inequality
• Gaussian moat
• Twin prime

• Guy, Richard K. (2004). Unsolved problems in number theory (3rded.). Springer-Verlag. ISBN978-0-387-20860-2. Zbl.

Further reading

• 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:. doi:. S2CID. Zbl.
• Mihăilescu, Preda (June 2014). (PDF). EMS Newsletter (92): 13–16. doi:. hdl:. ISSN.

External links

• Thomas R. Nicely, . This reference web site includes a list of all first known occurrence prime gaps.
• Weisstein, Eric W. . MathWorld.
• . PlanetMath.
• Armin Shams, , does not involve an 'arbitrarily big' constant as some other reported results.
• Chris Caldwell, ; an elementary introduction
• Andrew Granville, ; overview of the results obtained so far up to and including James Maynard's work of November 2013.
• Birke Heeren, Here you find a paper on how to calculate large gaps.