.
Prime gap
A prime gap is the difference between two successive prime numbers. The nth prime gap, denoted gn or g(pn) is the difference between the (n + 1)th and the nth prime numbers, i.e.
\( g_n = p_{n + 1}  p_n.\ \)
We have g1 = 1, g2 = g3 = 2, and g4 = 4. The sequence (gn) 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 OEIS).
By the definition of gn the following sum can be stated as
\( p_{n+1} = 2 + \sum_{i=1}^n g_i . \)
Simple observations
The first, smallest, and only odd prime gap is 1 between the only even prime number, 2, and the first odd prime, 3. All other prime gaps are even. There is only one pair of gaps between three consecutive odd natural numbers for which all are prime. These gaps are g_{2} and g_{3} between the primes 3, 5, and 7.
For any prime number P, we write P# for P primorial, that is, the product of all prime numbers up to and including P. If Q is the prime number following P, then the sequence
\( P\#+2, P\#+3,\ldots,P\#+(Q1) \)In the opposite direction, the twin prime conjecture asserts that gn = 2 for infinitely many integers n.
is a sequence of Q − 2 consecutive composite integers, so here there is a prime gap of at least length Q − 1. Therefore, there exist gaps between primes which are arbitrarily large, i.e., for any prime number P, there is an integer n with g_{n} ≥ P. (This is seen by choosing n so that p_{n} is the greatest prime number less than P# + 2.) Another way to see that arbitrarily large prime gaps must exist is the fact that the density of primes approaches zero, according to the prime number theorem. In fact, by this theorem, P# is very roughly a number the size of exp(P), and near exp(P) the average distance between consecutive primes is P.
In reality, prime gaps of P numbers can occur at numbers much smaller than P#. For instance, the smallest sequence of 71 consecutive composite numbers occurs between 31398 and 31468, whereas 71# has twentyseven digits – its full decimal expansion being 557940830126698960967415390.
Although the average gap between primes increases as the natural logarithm of the integer, the ratio of the maximum prime gap to the integers involved also increases as larger and larger numbers and gaps are encountered.
In the opposite direction, the twin prime conjecture asserts that g_{n} = 2 for infinitely many integers n.
Numerical results
Prime gap function
As of 2014 the largest known prime gap with identified probable prime gap ends has length 3311852, with 97953digit probable primes found by M. Jansen and J. K. Andersen.[1][2] The largest known prime gap with identified proven primes as gap ends has length 1113106, with 18662digit primes found by P. Cami, M. Jansen and J. K. Andersen.[1][3]
We say that gn is a maximal gap if gm < gn for all m < n. As of June 2014 the largest known maximal gap has length 1476, found by Tomás Oliveira e Silva. It is the 75th maximal gap, and it occurs after the prime 1425172824437699411.[4] Other record maximal gap terms can be found at OEIS A002386.
Usually the ratio of gn / ln(pn) is called the merit of the gap gn . In 1931, E. Westzynthius proved that prime gaps grow more than logarithmically. That is,[5]
\( \limsup_{n\to\infty}\frac{g_n}{\log p_n}=\infty. \)
As of January 2012, the largest known merit value, as discovered by M. Jansen, is 66520 / ln(1931*1933#/7230  30244) ≈ 35.4244594 where 1933# indicates the primorial of 1933. This number, 1931*1933#/7230  30244, is an 816digit prime. The next largest known merit value is 1476 / ln(1425172824437699411) ≈ 35.31.[1][6] This prime with the gap of 1476 is also the 75th maximal gap (the last one in the table below). Other record merit terms can be found at OEIS A111870.
The CramerShanksGranville ratio is the ratio of \( gn / (ln(pn))^2 \) .[6] The greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. Other record terms can be found at OEIS A111943.
The first 75 maximal gaps (n is not listed)



Further results
Upper bounds
Bertrand's postulate 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 says that the "average length" of the gap between a prime p and the next prime is ln p. The actual length of the gap might be much more or less than this. However, from the prime number theorem one can also deduce an upper bound on the length of prime gaps: for every ε > 0, there is a number N such that g_{n} < εp_{n} for all n > N.
One can deduce that the gaps get arbitrarily smaller in proportion to the primes: the quotient
\( \lim_{n\to\infty}\frac{g_n}{p_n}=0. \)
Hoheisel was the first to show[7] that there exists a constant θ < 1 such that
\( \pi(x + x^\theta)  \pi(x) \sim \frac{x^\theta}{\log(x)}\text{ as }x\text{ tends to infinity,} \)
hence showing that
\( g_n<p_n^\theta,\, \)
for sufficiently large n.
Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,[8] and to θ = 3/4 + ε, for any ε > 0, by Chudakov.[9]
A major improvement is due to Ingham,[10] who showed that if
\( \zeta(1/2 + it)=O(t^c)\, \)
for some positive constant c, where O refers to the big O notation, then
\( \pi(x + x^\theta)  \pi(x) \sim \frac{x^\theta}{\log(x)} \)
for any θ > (1 + 4c)/(2 + 4c). Here, as usual, ζ denotes the Riemann zeta function and π the primecounting 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 n^{3} and (n + 1)^{3} if n is sufficiently large.^{[11]} 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 n^{2} 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 showed that one may choose θ = 7/12.^{[12]}
A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.^{[13]}
In 2005, Daniel Goldston, János Pintz and Cem Yıldırım proved that
\( \liminf_{n\to\infty}\frac{g_n}{\log p_n}=0 \)
and later improved it[14] to
\( \liminf_{n\to\infty}\frac{g_n}{\sqrt{\log p_n}(\log\log p_n)^2}<\infty. \)
In 2013, Yitang Zhang proved that
\( \liminf_{n\to\infty} g_n < 7\cdot 10^7, \)
meaning that there are infinitely many gaps that do not exceed 70 million.[15] A Polymath Project collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013.[16] In November 2013, James Maynard introduced a new refinement of the GPY sieve, allowing him to reduce the bound to 600 and show that for any m there exists a bounded interval containing m prime numbers.[17] Using Maynard's ideas, the Polymath project has since improved the bound to 246.[18][16] Further, assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that N has been reduced to 12 and 6, respectively.[16]
Lower bounds
Robert Rankin, improving results by Erik Westzynthius and Paul Erdős, proved the existence of a constant c > 0 such that the inequality
\( g_n > \frac{c\log n\log\log n\log\log\log\log n}{(\log\log\log n)^2} \)
holds for infinitely many values n: he showed that one can take any constant c < eγ, where γ is the Euler–Mascheroni constant. The value of the constant c was later improved to any constant c < 2eγ.[19]
Paul Erdős offered a $5,000 prize for a proof or disproof that the constant c in the above inequality may be taken arbitrarily large.[20] This was proved independently by FordGreenKonyaginTao and James Maynard, in the positive, by two papers respectively sent to arXiv in 2014.[21][22]
The result was further improved to
\( g_n \gg \frac{\log n\log\log n\log\log\log\log n}{\log\log\log n} \)
(for infinitely many values of n) by FordGreenKonyaginMaynardTao.[23]
Conjectures about gaps between primes
Even better results are possible if it is assumed that the Riemann hypothesis is true. Harald Cramér proved that, under this assumption, the gap gn satisfies
\( g_n = O(\sqrt{p_n} \ln p_n), \)
using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that
\( g_n = O\left((\ln p_n)^2\right). \)
At the moment, the numerical evidence seems to point in this direction. See Cramér's conjecture for more details.
Firoozbakht's conjecture states that \( p_{n}^{1/n}\, \) (where p_n\, \)is the nth prime) is a strictly decreasing function of n, i.e.,
\( p_{n+1}^{1/(n+1)} < p_n^{1/n} \text{ for all } n \ge 1. \)
If this conjecture is true, then the function \( g_n = p_{n+1}  p_n \) satisfies \( g_n < (\log p_n)^2  \log p_n \text{ for all } n > 4 \). [24]
This is one of the strongest upper bound ever conjectured for prime gaps. Moreover, this conjecture implies Cramér's conjecture in a strong form and would be consistent with Daniel Shanks conjectured asymptotic equality of record gaps.[25]
By using tables of maximal gaps, Firoozbakht's conjecture has been verified for all primes below 4×10^{18}.[26]
Mean while, the Oppermann's conjecture is a conjecture which is weaker than Cramér's conjecture. The expected gap size with Oppermann's conjecture is
\( g_n < \sqrt{p_n}\, . \)
Andrica's conjecture, which is a weaker conjecture to Oppermann's, states that[20]
\( g_n < 2\sqrt{p_n} + 1.\, \)
This is a slight strengthening of Legendre's conjecture that between successive square numbers there is always a prime.
As an arithmetic function
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.^{[20]} The function is neither multiplicative nor additive.
See also
Portal icon Mathematics portal
Bonse's inequality
Gaussian moat
Twin prime
References
Andersen, Jens Kruse. "The Top20 Prime Gaps". Retrieved 20140613.
Largest known prime gap
A proven prime gap of 1113106
Maximal Prime Gaps
Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind", Commentationes PhysicoMathematicae Helsingsfors (in German) 5: 1–37, JFM 57.0186.02, Zbl 0003.24601.
NEW PRIME GAP OF MAXIMUM KNOWN MERIT
Hoheisel, G. (1930). "Primzahlprobleme in der Analysis". Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 33: 3–11. JFM 56.0172.02.
Heilbronn, H. A. (1933). "Über den Primzahlsatz von Herrn Hoheisel". Mathematische Zeitschrift 36 (1): 394–423. doi:10.1007/BF01188631.
Tchudakoff, N. G. (1936). "On the difference between two neighboring prime numbers". Math. Sb. 1: 799–814.
Ingham, A. E. (1937). "On the difference between consecutive primes". Quarterly Journal of Mathematics. Oxford Series 8 (1): 255–266. doi:10.1093/qmath/os8.1.255.
Cheng, YuanYou FuRui (2010). "Explicit estimate on primes between consecutive cubes". Rocky Mt. J. Math. 40: 117–153. doi:10.1216/rmj2010401117. Zbl 1201.11111.
Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones Mathematicae 15 (2): 164–170. doi:10.1007/BF01418933.
Baker, R. C.; Harman, G.; Pintz, J. (2001). "The difference between consecutive primes, II". Proceedings of the London Mathematical Society 83 (3): 532–562. doi:10.1112/plms/83.3.532.
"Primes in Tuples II". ArXiv. Retrieved 20131123.
Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761.
"Bounded gaps between primes". Polymath. Retrieved 20130721.
Maynard, James (2015). "Small gaps between primes". Annals of Mathematics 181 (1): 383–413. doi:10.4007/annals.2015.181.1.7. MR 3272929.
DHJ Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences 1 (12). doi:10.1186/s4068701400127.
Pintz, J. (1997). "Very large gaps between consecutive primes". J. Number Theory 63 (2): 286–301. doi:10.1006/jnth.1997.2081.
Guy (2004) §A8
Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao (2014) "Large gaps between consecutive prime numbers"
James Maynard (2014) "Large gaps between primes"
Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2015). "Long gaps between primes". arXiv:1412.5029.
Sinha, Nilotpal Kanti (2010), "On a new property of primes that leads to a generalization of Cramer's conjecture", arXiv.org > math > arXiv:1010.1399: 1–10.
Shanks, Daniel (1964), "On Maximal Gaps between Successive Primes", Mathematics of Computation (American Mathematical Society) 18 (88): 646–651, doi:10.2307/2002951, JSTOR 2002951, Zbl 0128.04203.
Kourbatov, Alexei. "prime Gaps: Firoozbakht Conjecture".
Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). SpringerVerlag. ISBN 9780387208602. Zbl 1058.11001.
Feliksiak, Jan (2013). The Symphony Of Primes, Distribution Of Primes And Riemann's Hypothesis. Xlibris. ISBN 9781479765584.
Further reading
Soundararajan, Kannan (2007). "Small gaps between prime numbers: the work of GoldstonPintzYıldırım". Bull. Am. Math. Soc., New Ser. 44 (1): 1–18. doi:10.1090/s0273097906011426. Zbl 1193.11086.
Mihăilescu, Preda (June 2014). "On some conjectures in additive number theory" (PDF). Newsletter of the European Mathematical Society (92): 13–16. doi:10.4171/NEWS. ISSN 1027488X.
External links
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 at PlanetMath.org.
Armin Shams, Reextending 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
www.primegaps.com A study of the gaps between consecutive prime numbers
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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