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. As of April 2022, the largest known prime gap with identified probable prime gap ends has length 7186572, with 208095-digit probable primes and merit M = 14.9985, found by Michiel Jansen using a sieve program developed by J. K. Andersen.[3][4] 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.[5][6]

As of September 2022, 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.[7][8]

The Cramér–Shanks–Granville ratio is the ratio of g_n / (ln(p_n))².[7] 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 August 2018 the largest known maximal prime gap has length 1550, found by Bertil Nyman. It is the 80th maximal gap, and it occurs after the prime 18361375334787046697.[12] 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[13] (see table below).

Prime gap function

Even better results are possible under the Riemann hypothesis. Harald Cramér proved[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]) Later, he conjectured that the gaps are even smaller. Roughly speaking, Cramér's conjecture states that

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

Firoozbakht's conjecture states that ${\displaystyle p_{n}^{1/n}\!}$ (where ${\displaystyle p_{n}}$ is the nth prime) 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 is true, then the function ${\displaystyle g_{n}=p_{n+1}-p_{n}}$ satisfies ${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}{\text{ for all }}n>4.}$[35] It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz[36][37][38] which suggest that ${\displaystyle g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}}$ infinitely often for any ${\displaystyle \varepsilon >0,}$ where ${\displaystyle \gamma }$ 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

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

As a result, under Oppermann's conjecture there exists ${\displaystyle m}$ (probably ${\displaystyle m=30}$) for which every natural number ${\displaystyle n>m}$ satisfies ${\displaystyle g_{n}<{\sqrt {p_{n}}}.}$

Andrica's conjecture, which is a weaker conjecture than Oppermann's, states that[39]

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

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 Zhang Yitang's result proves that it is true for at least one (currently unknown) value of k which is smaller than 70,000,000; as discussed above, this upper bound has been improved to 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.[39] 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.