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Prime gap - Wikipedia

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Difference between two successive prime numbers

Prime gap frequency distribution for primes up to 1.6 billion. Peaks occur at multiples of 6.^[1]

A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted g_n or g(p_n) is the difference between the (n + 1)st and the n-th prime numbers, i.e.,

g_n = p_n+1 − p_n.

We have g₁ = 1, g₂ = g₃ = 2, and g₄ = 4. The sequence (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, ... (sequence A001223 in the OEIS).

By the definition of 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[edit]

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

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. 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 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 16045848, with 385713-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 1113106 and merit 25.90, with 18662-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 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227. The prime gap between it and the next prime is 8350.^[6]^[7]

Largest known merit values (as of October 2020^[update])^[6]^[8]^[9]^[10] Merit g_n digits p_n Date Discoverer 41.938784 08350 0087 see above 2017 Gapcoin 39.620154 15900 0175 3483347771 × 409#/0030 − 7016 2017 Dana Jacobsen 38.066960 18306 0209 0650094367 × 491#/2310 − 8936 2017 Dana Jacobsen 38.047893 35308 0404 0100054841 × 953#/0210 − 9670 2020 Seth Troisi 37.824126 08382 0097 0512950801 × 229#/5610 − 4138 2018 Dana Jacobsen

The Cramér–Shanks–Granville ratio is the ratio 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. For comparison, the gap discovered by the Gapcoin network (whis Merit 41.938784), will only receive a value of 0.205879136 in this index. 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 2 ln n terms.^[12]

The 83 known maximal prime gaps Gaps 1 to 28 # g_n p_n 1 1 2 2 2 3 3 4 7 4 6 23 5 8 89 6 14 113 7 18 523 8 20 887 9 22 1129 10 34 1327 11 36 9551 12 44 15683 13 52 19609 14 72 31397 15 86 155921 16 96 360653 17 112 370261 18 114 492113 19 118 1349533 20 132 1357201 21 148 2010733 22 154 4652353 23 180 17051707 24 210 20831323 25 220 47326693 26 222 122164747 27 234 189695659 28 248 191912783 Gaps 29 to 56 # g_n p_n 29 250 387096133 30 282 436273009 31 288 1294268491 32 292 1453168141 33 320 2300942549 34 336 3842610773 35 354 4302407359 36 382 10726904659 37 384 20678048297 38 394 22367084959 39 456 25056082087 40 464 42652618343 41 468 127976334671 42 474 182226896239 43 486 241160624143 44 490 297501075799 45 500 303371455241 46 514 304599508537 47 516 416608695821 48 532 461690510011 49 534 614487453523 50 540 738832927927 51 582 1346294310749 52 588 1408695493609 53 602 1968188556461 54 652 2614941710599 55 674 7177162611713 56 716 13829048559701 Gaps 57 to 83 # g_n p_n 57 766 19581334192423 58 778 42842283925351 59 804 90874329411493 60 806 171231342420521 61 906 218209405436543 62 916 1189459969825483 63 924 1686994940955803 64 1,132 1693182318746371 65 1,184 43841547845541059 66 1,198 55350776431903243 67 1,220 80873624627234849 68 1,224 203986478517455989 69 1,248 218034721194214273 70 1,272 305405826521087869 71 1,328 352521223451364323 72 1,356 401429925999153707 73 1,370 418032645936712127 74 1,442 804212830686677669 75 1,476 1425172824437699411 76 1,488 5733241593241196731 77 1,510 6787988999657777797 78 1,526 15570628755536096243 79 1,530 17678654157568189057 80 1,550 18361375334787046697 81 1,552 18470057946260698231 82 1,572 18571673432051830099 83 1,676 20733746510561442863

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

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, there is a number N such that for all n > N,

g n < p n ϵ {\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

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

g n < p n θ {\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 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.

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 = 0.583.^[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

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^[20] 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.^[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 also show that the gaps between primes m apart are bounded for all m. That is, for any m there exists a bound Δ_m such that p_n+m − p_n ≤ Δ_m for infinitely many n.^[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]

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 < 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

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.^[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]

Conjectures about gaps between primes[edit]

As described above, the best proven bound on gap sizes is g_n < p_n^0.525 (for n sufficiently large; we do not worry about 5 − 3 > 3^0.525 or 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 θ = 0.5.

Legendre's conjecture that there always exists a prime between successive square numbers implies that g_n = O(√p_n). Andrica's conjecture states that^[32]

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

Oppermann's conjecture makes the stronger claim that, for sufficiently large n (probably n > 30),

g n < p n . {\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

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 infinitesimally larger exponent.^[34])

Prime gap function

In the same article, he 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.

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 p_n^1/n is a strictly decreasing function of n, i.e.,

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

If this conjecture were true, then g_n < (log p_n)² − log p_n − 1 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 g_n > (2 − ϵ)e^−γ(log p_n)² > (1.1229 − ϵ)(log p_n)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 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.

As an arithmetic function[edit]

The gap g_n between the nth and (n + 1)th 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.

Mathematics portal

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