Prime Gaps

A prime gap of length is a run of consecutive composite numbers between two successive primes, where is a positive integer. Therefore, the difference between two successive primes and bounding a prime gap of length is , where is the th prime number. Since the prime difference function

(1)

is always even (except for ), all prime gaps are also even. The notation is commonly used to denote the smallest prime corresponding to the start of a prime gap of length , i.e., such that is prime, , , ..., are all composite, and is prime (with the additional constraint that no smaller number satisfying these properties exists).

The maximal prime gap is the length of the largest prime gap that begins with a prime less than some maximum value . For , 2, ..., is given by 4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ... (OEIS A053303).

Arbitrarily large prime gaps exist. For example, for any , the numbers , , ..., are all composite (Havil 2003, p. 170). However, no general method more sophisticated than an exhaustive search is known for the determination of first occurrences and maximal prime gaps (Nicely 1999).

Writing for the -fold nested logarithm, Ford et al. (2018) proved the asymptotic lower bound

(2)

GPT 5.6 Sol (2026) claimed the stronger bound

(3)

an improvement by a factor of . The paper identifies GPT 5.6 Sol as its author. Alexeev (2026) formalized the two main statements unconditionally in Lean. After discussions with Terry Tao and James Maynard, Green (2026) reported that he was "more or less convinced" that the result was correct. The manuscript has not been refereed, and the correspondence between the manuscript and formalization has not received independent review (VibeMathed 2026).

Cramér (1937) and Shanks (1964) conjectured that

(4)

Wolf conjectures a slightly different form

(5)

which agrees better with numerical evidence.

Wolf conjectures that the maximal gap between two consecutive primes less than appears approximately at

(6)

where is the prime counting function and is the twin primes constant. Setting reduces to Cramer's conjecture for large ,

(7)

It is known that there is a prime gap of length 804 following , and a prime gap of length following (Baugh and O'Hara 1992). H. Dubner (2001) discovered a prime gap of length between two 3396-digit probable primes. On Jan. 15, 2004, J. K. Andersen and H. Rosenthal found a prime gap of length between two probable primes of digits each. In January-May 2004, Hans Rosenthal and Jens Kruse Andersen found a prime gap of length between two probable primes with digits each (Andersen 2004).

The average prime gap through is for and is asymptotic to by the prime number theorem. This average grows without bound even though bounded gaps occur infinitely often. The merit of an individual prime gap compares it to this local average scale, and is given by . As of Oct. 2026, the largest known merit for a gap between proven primes is 42.5192, attained by a gap of length 59724 following a 611-digit prime number and discovered by M. Vojvoda (Prime Gap List Project 2026, Vojvoda 2026). A larger merit of 42.5402 is listed for a gap of length 14282 between 146-digit probable primes discovered by S. Troisi (Prime Gap List Project 2026). The primes preceding gaps of increasing record merit begin 2, 3, 7, 113, 1129, 1327, 19609, ... (OEIS A111870).

Young and Potler (1989) determined the first occurrences of prime gaps up to , with all first occurrences found between 1 and 673. Nicely (1999) has extended the list of maximal prime gaps. The following table gives the values of for record gaps, namely gaps longer than every preceding prime gap (OEIS A005250 and A002386). As of Oct. 16, 2026, the confirmed records extend through a gap of length 1854 following (Andersen and Luhn 2026). First occurrences of lengths that do not set a record, such as the gap of length 10 following 139, are omitted.

1	2	490
2	3	500
4	7	514
6	23	516
8	89	532
14	113	534
18	523	540
20	887	582
22		588
34		602
36		652
44		674
52		716
72		766
86		778
96		804
112		806
114		906
118		916
132		924
148
154
180
210
220
222
234
248
250
282
288
292
320
336
354
382
384
394
456
464
468
474
486

Define

(8)

as the infimum limit of the ratio of the th prime difference to the natural logarithm of the th prime number. If there are an infinite number of twin primes, then . This follows since it must then be true that infinitely often, say at for , 2, ..., so a necessary condition for the twin prime conjecture to hold is that

			(9)
			(10)
			(11)
			(12)

However, this condition is not sufficient, since the same proof works if 2 is replaced by any constant.

Hardy and Littlewood showed in 1926 that, subject to the truth of the generalized Riemann hypothesis, . This was subsequently improved by Rankin (again assuming the generalized Riemann hypothesis) to . In 1940, Erdős used sieve theory to show for the first time with no assumptions that . This was subsequently improved to 15/16 (Ricci), (Bombieri and Davenport 1966), and (Pil'Tai 1972), as quoted in Le Lionnais (1983, p. 26). Huxley (1973, 1977) obtained , which was improved by Maier in 1986 to , which was the best result known until 2003 (American Institute of Mathematics).

At a March 2003 meeting on elementary and analytic number theory in Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof that . While the original proof turned out to be flawed (Mackenzie 2003ab), the result has now been established by a new proof (American Institute of Mathematics 2005, Cipra 2005, Devlin 2005, Goldston et al. 2005ab).

Zhang (2014) subsequently proved the much stronger unnormalized result that the infimum limit is finite, initially obtaining . Work of Maynard (2015) and the Polymath8 project then reduced the unconditional bound to (Polymath 2014). Thus infinitely many pairs of consecutive primes differ by at most 246. This result on bounded gaps between primes does not prove that a gap of exactly 2 occurs infinitely often, so it does not prove the twin prime conjecture.

Assuming the Elliott-Halberstam conjecture gives the conditional bound (Maynard 2015). It also gives (Polymath 2014), improving Maynard's earlier bound . The stronger generalized conjecture gives and (Polymath 2014). Neither conditional result proves that .

OpenAI (2026a) reported the stronger maximal-gap lower bound

(13)

where again denotes the -fold nested natural logarithm. GPT-6 Astra discovered the proof. VibeMathed's September verification replayed the Lean proof and checked its axioms, but independent mathematical review of the new argument had not been reported as of Sep. 7, 2026.

Mossinghoff and Trudgian (2026) call a consecutive-prime gap frangible when every integer strictly between its endpoints has a least prime factor strictly less than the gap . This is the property in Erdős problem 682. Assuming the Dickson-Hardy-Littlewood prime-tuples conjecture, the number of such gaps with left endpoint satisfies

(14)

where . The constant bounds are rigorous, while the counting asymptotic is conditional (Mossinghoff and Trudgian 2026).