Prime gap

The sequence (gn) of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered.

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 gm ≥ N. However, prime gaps of n numbers can occur at numbers much smaller than n!.

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.

[2] In the opposite direction, the twin prime conjecture posits that gn = 2 for infinitely many integers n. Usually the ratio of

We say that gn is a maximal gap, if gm < gn for all m < n. As of October 2024[update], the largest known maximal prime gap has length 1676, found by Brian Kehrig.

The sequence of maximal gaps up to the nth prime is conjectured to have about

Hoheisel (1930) was the first to show[13] a sublinear dependence; that there exists a constant θ < 1 such that hence showing that for sufficiently large n. Hoheisel obtained the possible value 32999/33000 for θ.

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

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

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 and 2 years later improved this[20] to In 2013, Yitang Zhang proved that meaning that there are infinitely many gaps that do not exceed 70 million.

[22] In 1931, Erik Westzynthius proved that maximal prime gaps grow more than logarithmically.

That is,[2] In 1938, Robert Rankin proved the existence of a constant c > 0 such that the inequality holds for infinitely many values of n, improving the results of Westzynthius and Paul Erdős.

[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.

[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.

), but it is observed that even maximal gaps are significantly smaller than that, leading to a plethora of unproven conjectures.

Legendre's conjecture that there always exists a prime between successive square numbers implies that

Harald Cramér came close, proving[33] that the Riemann hypothesis implies the gap gn satisfies using the big O notation.

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

Roughly speaking, Cramér's conjecture states that a polylogarithmic growth rate slower than any exponent

As this matches the observed growth rate of prime gaps, there are a number of similar conjectures.

[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

Polignac's conjecture states that every positive even number k occurs as a prime gap infinitely often.

The gap gn between the nth and (n + 1)st prime numbers is an example of an arithmetic function.

In this context it is usually denoted dn and called the prime difference function.