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

: $n!+2,\backslash ;\; n!+3,\backslash ;\; \backslash ldots,\backslash ;\; n!+n,$

the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of {{math|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 {{math|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 {{math|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.{{Citation |last=Westzynthius |first=E. |title=Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind |language=de|journal=Commentationes Physico-Mathematicae Helsingsfors |volume=5 |year=1931 |pages=1–37 | zbl=0003.24601 | jfm=57.0186.02 }}.

In the opposite direction, the twin prime conjecture posits that {{nowrap|1=g_n = 2}} for infinitely many integers n.

Usually the ratio of $\backslash frac\{g\_n\}\{\backslash ln(p\_n)\}$ 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 16,045,848, with 385,713-digit probable primes and merit M = 18.067, found by Andreas Höglund in {{date|March 2024}}.{{Cite web |last=ATH |date=2024-03-11 |title=Announcement at Mersenneforum.org |url=https://mersenneforum.org/showpost.php?p=652565&postcount=300 |url-status=live |archive-url=https://web.archive.org/web/20240312154958/https://mersenneforum.org/showpost.php?p=652565&postcount=300 |archive-date=2024-03-12 |website=Mersenneforum.org}} The largest known prime gap with identified proven primes as gap ends has length 1,113,106 and merit 25.90, with 18,662-digit primes found by P. Cami, M. Jansen and J. K. Andersen.{{cite web|last1=Andersen|first1=Jens Kruse|title=The Top-20 Prime Gaps|url=http://primerecords.dk/primegaps/gaps20.htm|access-date=2014-06-13|archive-date=December 27, 2019|archive-url=https://web.archive.org/web/20191227185818/http://primerecords.dk/primegaps/gaps20.htm|url-status=live}}{{Cite web |last=Andersen |first=Jens Kruse |date=8 March 2013 |title=A megagap with merit 25.9 |url=http://primerecords.dk/primegaps/gap1113106.htm |access-date=2022-09-29 |website=primerecords.dk |archive-date=December 25, 2019 |archive-url=https://web.archive.org/web/20191225142708/http://primerecords.dk/primegaps/gap1113106.htm |url-status=live }}

{{As of|2022|09}}, 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 {{zwsp|2|9|3|7|0|3|2|3|4|0|6|8|0|2|2|5|9|0|1|5|8|7|2|3|7|6|6|1|0|4|4|1|9|4|6|3|4|2|5|7|0|9|0|7|5|5|7|4|8|1|1|7|6|2|0|9|8|5|8|8|7|9|8|2|1|7|8|9|5|7|2|8||8|5|8|6|7|6|7|2|8|1|4|3|2|2|7}}. The prime gap between it and the next prime is 8350.{{Cite web |last=Nicely |first=Thomas R. |date=2019 |title=NEW PRIME GAP OF MAXIMUM KNOWN MERIT |url=https://faculty.lynchburg.edu/~nicely/#MaxMerit |access-date=2022-09-29 |website=faculty.lynchburg.edu |archive-date=April 30, 2021 |archive-url=https://web.archive.org/web/20210430231853/https://faculty.lynchburg.edu/~nicely/#MaxMerit |url-status=live }}{{cite web | url=https://github.com/primegap-list-project/prime-gap-list | title=Prime Gap Records | website=GitHub | date=June 11, 2022 }}

The Cramér–Shanks–Granville ratio is the ratio of g_n / (ln(p_n))². 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 {{OEIS2C|id=A111943}}.

We say that g_n is a maximal gap, if g_m < g_n for all m < n. {{As of|2024|10}}, 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.{{cite web|last=Andersen|first=Jens Kruse|title=Record prime gaps|url=https://www.pzktupel.de/JensKruseAndersen/risinggap.php|access-date=Oct 10, 2024}} Other record (maximal) gap sizes can be found in {{OEIS2C|id=A005250}}, with the corresponding primes p_n in {{OEIS2C|id=A002386}}, and the values of n in {{OEIS2C|id=A005669}}. The sequence of maximal gaps up to the nth prime is conjectured to have about $2\backslash ln\; n$ terms.{{cite journal |last1=Kourbatov |first1=A. |last2=Wolf |first2=M. |title=On the first occurrences of gaps between primes in a residue class |journal=Journal of Integer Sequences |volume=23 |issue=Article 20.9.3 |url=https://cs.uwaterloo.ca/journals/JIS/VOL23/Wolf/wolf2.html |year=2020 |arxiv=2002.02115 |mr=4167933 |zbl=1444.11191 |s2cid=211043720 |access-date=December 3, 2020 |archive-date=April 12, 2021 |archive-url=https://web.archive.org/web/20210412194750/https://cs.uwaterloo.ca/journals/JIS/VOL23/Wolf/wolf2.html |url-status=live }}

|

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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

:$\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{g\_n\}\{p\_n\}=0.$

In other words (by definition of a limit), for every $\backslash epsilon\; >\; 0$, there is a number $N$ such that for all $n\; >\; N$

:.

Hoheisel (1930) was the first to show{{cite journal |first=G. |last=Hoheisel |title=Primzahlprobleme in der Analysis |journal=Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin |volume=33 |pages=3–11 |year=1930 | jfm=56.0172.02 }} a sublinear dependence; that there exists a constant θ < 1 such that

:$\backslash pi(x\; +\; x^\backslash theta)\; -\; \backslash pi(x)\; \backslash sim\; \backslash frac\{x^\backslash theta\}\{\backslash log(x)\}\; \backslash text\{\; as\; \}\; x\; \backslash to\; \backslash infty,$

hence showing that

:

for sufficiently large n.

Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,{{cite journal |first=H. A. |last=Heilbronn |title=Über den Primzahlsatz von Herrn Hoheisel |journal=Mathematische Zeitschrift |volume=36 |issue=1 |pages=394–423 |year=1933 |doi=10.1007/BF01188631 |jfm=59.0947.01 |s2cid=123216472 }} and to θ = 3/4 + ε, for any ε > 0, by Chudakov.{{cite journal |first=N. G. |last=Tchudakoff |title=On the difference between two neighboring prime numbers |journal=Mat. Sb. |volume=1 |pages=799–814 |year=1936 |zbl=0016.15502}}

A major improvement is due to Ingham,{{cite journal |last=Ingham |first=A. E. |title=On the difference between consecutive primes |journal=Quarterly Journal of Mathematics |series=Oxford Series |volume=8 |issue=1 |pages=255–266 |year=1937 |doi=10.1093/qmath/os-8.1.255 |bibcode=1937QJMat...8..255I }} who showed that for some positive constant c,

:if $\backslash zeta(1/2\; +\; it)\; =\; O(t^c)$ then $\backslash pi(x\; +\; x^\backslash theta)\; -\; \backslash pi(x)\; \backslash sim\; \backslash frac\{x^\backslash theta\}\{\backslash log(x)\}$ for any $\backslash theta\; >\; (1\; +\; 4c)/(2\; +\; 4c).$

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.{{cite journal | last=Cheng | first=Yuan-You Fu-Rui | title=Explicit estimate on primes between consecutive cubes | zbl=1201.11111 | journal=Rocky Mt. J. Math. | volume=40 | pages=117–153 | year=2010 | doi=10.1216/rmj-2010-40-1-117| arxiv=0810.2113 | s2cid=15502941 }} 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 = {{overline|0.58|3}}.{{cite journal |last=Huxley |first=M. N. |year=1972 |title=On the Difference between Consecutive Primes |journal=Inventiones Mathematicae |volume=15 |issue=2 |pages=164–170 |doi=10.1007/BF01418933 |bibcode=1971InMat..15..164H |s2cid=121217000 }}

A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.{{cite journal |last1=Baker |first1=R. C. |first2=G. |last2=Harman |first3=J. |last3=Pintz |year=2001 |title=The difference between consecutive primes, II |journal=Proceedings of the London Mathematical Society |volume=83 |issue=3 |pages=532–562 |doi=10.1112/plms/83.3.532 |s2cid=8964027 |citeseerx=10.1.1.360.3671 |url=https://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf }}

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

:$\backslash liminf\_\{n\backslash to\backslash infty\}\backslash frac\{g\_n\}\{\backslash log\; p\_n\}\; =\; 0$

and 2 years later improved this{{cite journal

| arxiv=0710.2728

| title=Primes in Tuples II

| last1= Goldston| first1=Daniel A.

| last2= Pintz| first2=János

| last3= Yıldırım| first3=Cem Yalçin

| doi=10.1007/s11511-010-0044-9

| journal=Acta Mathematica

| volume=204

| issue=1

| pages=1–47

| year=2010| s2cid=7993099

}} to

:

In 2013, Yitang Zhang proved that

:

meaning that there are infinitely many gaps that do not exceed 70 million.{{cite journal | title = Bounded gaps between primes | first = Yitang | last = Zhang | author-link=Yitang Zhang | journal = Annals of Mathematics | year=2014 | volume=179 | issue=3 | pages=1121–1174 | doi=10.4007/annals.2014.179.3.7 | mr=3171761| doi-access=free }} A Polymath Project collaborative effort to optimize Zhang's bound managed to lower the bound to 4680 on July 20, 2013.{{cite web|url=http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes|title=Bounded gaps between primes|publisher=Polymath|access-date=2013-07-21|archive-date=February 28, 2020|archive-url=https://web.archive.org/web/20200228120914/http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes|url-status=live}} 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.{{cite journal | title=Small gaps between primes | last=Maynard | first=James | author-link=James Maynard (mathematician) | date=January 2015 | journal=Annals of Mathematics | volume=181 | issue=1 | pages=383–413 | mr=3272929 | doi=10.4007/annals.2015.181.1.7 | doi-access=free | arxiv=1311.4600 | s2cid=55175056 }} Using Maynard's ideas, the Polymath project improved the bound to 246;{{cite journal | author=D.H.J. Polymath | title=Variants of the Selberg sieve, and bounded intervals containing many primes | journal=Research in the Mathematical Sciences | volume=1 | number=12 | doi=10.1186/s40687-014-0012-7 | arxiv=1407.4897 | year=2014 | mr=3373710| s2cid=119699189 | doi-access=free }} assuming the Elliott–Halberstam conjecture and its generalized form, the bound has been reduced to 12 and 6, respectively.

In 1931, Erik Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,

:$\backslash limsup\_\{n\backslash to\backslash infty\}\backslash frac\{g\_n\}\{\backslash log\; p\_n\}=\backslash infty.$

In 1938, Robert Rankin proved the existence of a constant c > 0 such that the inequality

:$g\_n\; >\; \backslash frac\{c\backslash \; \backslash log\; n\backslash \; \backslash log\backslash log\; n\backslash \; \backslash log\backslash log\backslash log\backslash log\; n\}\{(\backslash log\backslash log\backslash log\; n)^2\}$

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.{{cite journal |first=J. |last=Pintz |title=Very large gaps between consecutive primes |journal=J. Number Theory |volume=63 |issue=2 |pages=286–301 |year=1997 |doi=10.1006/jnth.1997.2081 |doi-access=free }}

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.{{cite book

|editor1-last= Erdős

|editor1-first= Paul

|editor2-last= Bollobás

|editor2-first= Béla

|editor3-last= Thomason

|editor3-first= Andrew

|date= 1997

|title= Combinatorics, Geometry and Probability: A Tribute to Paul Erdős

|url= https://books.google.com/books?id=1E6ZwSEtPAEC&pg=PA1

|page= 1

|publisher= Cambridge University Press

|isbn= 9780521584722

|access-date= September 29, 2022

|archive-date= September 29, 2022

|archive-url= https://web.archive.org/web/20220929060157/https://books.google.com/books?id=1E6ZwSEtPAEC&pg=PA1

|url-status= live

}} This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.{{cite journal | first1=Kevin | last1=Ford | first2=Ben | last2=Green | first3=Sergei | last3=Konyagin | first4=Terence | last4=Tao | year=2016 | arxiv=1408.4505 | title=Large gaps between consecutive prime numbers | journal=Ann. of Math. | volume=183 | issue=3 | pages=935–974 | doi=10.4007/annals.2016.183.3.4 | mr=3488740| s2cid=16336889 }}{{cite journal | first=James | last=Maynard | year=2016 | volume=183 | issue=3 | arxiv=1408.5110 | title=Large gaps between primes | pages=915–933 | doi=10.4007/annals.2016.183.3.3 | mr = 3488739 | journal=Ann. of Math.| s2cid=119247836 }}

The result was further improved to

:$g\_n\; >\; \backslash frac\{c\backslash \; \backslash log\; n\backslash \; \backslash log\backslash log\; n\backslash \; \backslash log\backslash log\backslash log\backslash log\; n\}\{\backslash log\backslash log\backslash log\; n\}$

for infinitely many values of n by Ford–Green–Konyagin–Maynard–Tao.{{cite journal | arxiv=1412.5029 | title=Long gaps between primes | last1=Ford | first1=Kevin | last2=Green | first2=Ben | last3=Konyagin | first3=Sergei | last4=Maynard | first4=James | last5=Tao | first5=Terence | year=2018| mr=3718451 | journal=J. Amer. Math. Soc. | volume=31 | issue=1 | pages=65–105 | doi=10.1090/jams/876 | s2cid=14487001 }}

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.{{Cite web |last=Tao |first=Terence |date=16 December 2014 |title=Long gaps between primes / What's new |url=https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/ |access-date=August 29, 2019 |archive-date=June 9, 2019 |archive-url=https://web.archive.org/web/20190609134631/https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/ |url-status=live }}

Lower bounds for chains of primes have also been determined.{{cite arXiv | first1=Kevin | last1=Ford | first2=James | last2=Maynard | first3=Terence | last3=Tao | title = Chains of large gaps between primes | eprint=1511.04468 | date=2015-10-13| class=math.NT }}

As described above, the best proven bound on gap sizes is (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 $\backslash theta\; =\; 0.5$.

Legendre's conjecture that there always exists a prime between successive square numbers implies that $g\_n\; =\; O(\backslash sqrt\{p\_n\})$. Andrica's conjecture states thatGuy (2004) §A8

:

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

:

All of these remain unproved. Harald Cramér came close, proving{{cite journal |last=Cramér |first=Harald |title=On the order of magnitude of the difference between consecutive prime numbers |journal=Acta Arithmetica |volume=2 |year=1936 |pages=23–46 |doi=10.4064/aa-2-1-23-46 |doi-access=free }} that the Riemann hypothesis implies the gap g_n satisfies

:$g\_n\; =\; O(\backslash sqrt\{p\_n\}\; \backslash log\; p\_n),$

using the big O notation. (In fact this result needs only the weaker Lindelöf hypothesis, if one can tolerate an infinitesimally larger exponent.{{cite journal |first=Albert E. |last=Ingham |author-link=Albert Ingham |title=On the difference between consecutive primes |journal=Quarterly Journal of Mathematics |location=Oxford |volume=8 |issue=1 |pages=255–266 |year=1937 |doi=10.1093/qmath/os-8.1.255 |bibcode=1937QJMat...8..255I |url=https://dustri.org/b/files/On_the_difference_between_consecutive_primes_-_A.E.Ingham.pdf |archive-url=https://web.archive.org/web/20221205135811/https://dustri.org/b/files/On_the_difference_between_consecutive_primes_-_A.E.Ingham.pdf |archive-date=2022-12-05 |url-status=live}})

File:Wikipedia primegaps.png

In the same article, he conjectured that the gaps are far smaller. Roughly speaking, Cramér's conjecture states that

:$g\_n\; =\; O\backslash !\backslash left((\backslash log\; p\_n)^2\backslash right)\backslash !,$

a polylogarithmic growth rate slower than any exponent $\backslash theta\; >\; 0$.

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 $p\_\{n\}^\{1/n\}\backslash !$ is a strictly decreasing function of n, i.e.,

:

If this conjecture were true, then {{cite arXiv |last=Sinha |first=Nilotpal Kanti |title=On a new property of primes that leads to a generalization of Cramer's conjecture |year=2010 |eprint=1010.1399 |class=math.NT }}{{cite journal |last=Kourbatov |first=Alexei |title=Upper bounds for prime gaps related to Firoozbakht's conjecture |journal=Journal of Integer Sequences |volume=18 |issue=11 |article-number=15.11.2 |year=2015 |arxiv=1506.03042 }} It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz{{cite journal |last=Granville |first=Andrew |author-link=Andrew Granville |title=Harald Cramér and the distribution of prime numbers |journal=Scandinavian Actuarial Journal |volume=1 |year=1995 |pages=12–28 |url=http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf |doi=10.1080/03461238.1995.10413946 |citeseerx=10.1.1.129.6847 |access-date=March 2, 2016 |archive-date=September 23, 2015 |archive-url=https://web.archive.org/web/20150923212842/http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf |url-status=live }}.{{cite book |last=Granville |first=Andrew |title=Proceedings of the International Congress of Mathematicians |chapter=Unexpected Irregularities in the Distribution of Prime Numbers |author-link=Andrew Granville |volume=1 |year=1995 |pages=388–399 |chapter-url=http://www.dms.umontreal.ca/~andrew/PDF/icm.pdf |doi=10.1007/978-3-0348-9078-6_32 |isbn=978-3-0348-9897-3 |access-date=March 2, 2016 |archive-date=May 7, 2016 |archive-url=https://web.archive.org/web/20160507093313/http://www.dms.umontreal.ca/~andrew/PDF/icm.pdf |url-status=live }}.{{cite journal|last=Pintz|first=János|author-link=János Pintz|title=Cramér vs. Cramér: On Cramér's probabilistic model for primes|journal=Functiones et Approximatio Commentarii Mathematici|volume=37|issue=2|date=September 2007|pages=232–471|doi=10.7169/facm/1229619660|doi-access=free}} which suggest that $g\_n\; >\; \backslash frac\{2-\backslash varepsilon\}\{e^\backslash gamma\}(\backslash log\; p\_n)^2\; >\; (1.1229-\backslash varepsilon)(\backslash log\; p\_n)^2$ infinitely often for any $\backslash varepsilon>0,$ where $\backslash gamma$ 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.

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. The function is neither multiplicative nor additive.

{{clear}}

{{Portal|Mathematics}}

• Bonse's inequality
• Gaussian moat
• Twin prime

{{reflist}}

• {{cite book |last=Guy | first=Richard K. | author-link=Richard K. Guy | title=Unsolved problems in number theory | publisher=Springer-Verlag |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 }}

• {{cite journal | last=Soundararajan | first=Kannan | author-link=Kannan Soundararajan | title=Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım | zbl=1193.11086 | journal=Bull. Am. Math. Soc. |series=New Series | volume=44 | number=1 | pages=1–18 | year=2007 | doi=10.1090/s0273-0979-06-01142-6| arxiv=math/0605696 | s2cid=119611838 }}
• {{cite journal | first=Preda | last=Mihăilescu | author-link=Preda Mihăilescu | journal=EMS Newsletter | number=92 | date=June 2014 | issn=1027-488X | title=On some conjectures in additive number theory | pages=13–16 | url=http://www.ems-ph.org/journals/newsletter/pdf/2014-06-92.pdf | doi=10.4171/NEWS | hdl=2117/17085 | hdl-access=free }}

• Thomas R. Nicely, [https://faculty.lynchburg.edu/~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.
• {{MathWorld|urlname=PrimeDifferenceFunction|title=Prime Difference Function}}
• {{planetmath reference|urlname=PrimeDifferenceFunction|title=Prime Difference Function}}
• Armin Shams, [https://link.springer.com/article/10.1007%2Fs11253-008-0034-7 Re-extending Chebyshev's theorem about Bertrand's conjecture], does not involve an 'arbitrarily big' constant as some other reported results.
• Chris Caldwell, [http://primes.utm.edu/notes/gaps.html Gaps Between Primes]; an elementary introduction
• Andrew Granville, [http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf Primes in Intervals of Bounded Length]; overview of the results obtained so far up to and including James Maynard's work of November 2013.
• Birke Heeren, [https://prime-numbers.de] Here you find the large prime number gaps and a paper on how to calculate such large gaps.

{{Prime number classes}}

Category:Arithmetic functions

Category:Prime numbers