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Prime gap

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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 gn or g(pn) is the difference between the (n + 1)-st and the n-th prime numbers, i.e.

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 the OEIS).

By the definition of gn every prime can be written as

Simple observations

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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

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 gmN.

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 ek; 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 gn = 2 for infinitely many integers n.

Numerical results

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Usually the ratio of is called the merit of the gap gn. Informally, the merit of a gap gn can be thought of as the ratio of the size of the gap compared to the average prime gap sizes in the vicinity of pn.

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 March 2024.[3] 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.[4][5]

As of September 2022, 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 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.[6][7]

Largest known merit values (as of October 2020)[6][8][9][10]
Merit gn digits pn 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 of gn / (ln(pn))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. Other record terms can be found at OEISA111943.

We say that gn is a maximal gap, if gm < gn for all m < n. As of October 2024, 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 OEISA005250, with the corresponding primes pn in OEISA002386, and the values of n in OEISA005669. The sequence of maximal gaps up to the nth prime is conjectured to have about terms.[12]

The 83 known maximal prime gaps
Gaps 1 to 28
# gn pn
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 1,129
10 34 1,327
11 36 9,551
12 44 15,683
13 52 19,609
14 72 31,397
15 86 155,921
16 96 360,653
17 112 370,261
18 114 492,113
19 118 1,349,533
20 132 1,357,201
21 148 2,010,733
22 154 4,652,353
23 180 17,051,707
24 210 20,831,323
25 220 47,326,693
26 222 122,164,747
27 234 189,695,659
28 248 191,912,783
Gaps 29 to 56
# gn pn
29 250 387,096,133
30 282 436,273,009
31 288 1,294,268,491
32 292 1,453,168,141
33 320 2,300,942,549
34 336 3,842,610,773
35 354 4,302,407,359
36 382 10,726,904,659
37 384 20,678,048,297
38 394 22,367,084,959
39 456 25,056,082,087
40 464 42,652,618,343
41 468 127,976,334,671
42 474 182,226,896,239
43 486 241,160,624,143
44 490 297,501,075,799
45 500 303,371,455,241
46 514 304,599,508,537
47 516 416,608,695,821
48 532 461,690,510,011
49 534 614,487,453,523
50 540 738,832,927,927
51 582 1,346,294,310,749
52 588 1,408,695,493,609
53 602 1,968,188,556,461
54 652 2,614,941,710,599
55 674 7,177,162,611,713
56 716 13,829,048,559,701
Gaps 57 to 83
# gn pn
57 766 19,581,334,192,423
58 778 42,842,283,925,351
59 804 90,874,329,411,493
60 806 171,231,342,420,521
61 906 218,209,405,436,543
62 916 1,189,459,969,825,483
63 924 1,686,994,940,955,803
64 1,132 1,693,182,318,746,371
65 1,184 43,841,547,845,541,059
66 1,198 55,350,776,431,903,243
67 1,220 80,873,624,627,234,849
68 1,224 203,986,478,517,455,989
69 1,248 218,034,721,194,214,273
70 1,272 305,405,826,521,087,869
71 1,328 352,521,223,451,364,323
72 1,356 401,429,925,999,153,707
73 1,370 418,032,645,936,712,127
74 1,442 804,212,830,686,677,669
75 1,476 1,425,172,824,437,699,411
76 1,488 5,733,241,593,241,196,731
77 1,510 6,787,988,999,657,777,797
78 1,526 15,570,628,755,536,096,243
79 1,530 17,678,654,157,568,189,057
80 1,550 18,361,375,334,787,046,697
81 1,552 18,470,057,946,260,698,231
82 1,572 18,571,673,432,051,830,099
83 1,676 20,733,746,510,561,442,863

Further results

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Upper bounds

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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 pn +1 < 2pn, which means gn < pn .

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

In other words (by definition of a limit), for every , there is a number such that for all

.

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. 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 then for any

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 = 0.58(3).[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

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.[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 show that for any m there exists a bounded interval with an infinite number of translations each of which containing m prime numbers[incomprehensible].[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]

Lower bounds

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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. He later showed that one can take any constant c < eγ, where γ is the Euler–Mascheroni constant. The value of the constant c was improved in 1997 to any value less than 2eγ.[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

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

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As described above, the best proven bound on gap sizes is (for sufficiently large; we do not worry about or ), 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 .

Legendre's conjecture that there always exists a prime between successive square numbers implies that . Andrica's conjecture states that[32]

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

All of these remain unproved. 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.[34])

Prime gap function

In the same article, he conjectured that the gaps are far smaller. 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. Firoozbakht's conjecture is slightly stronger, stating that is a strictly decreasing function of n, i.e.,

If this conjecture were true, then [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 infinitely often for any where 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

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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.[32] The function is neither multiplicative nor additive.

See also

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References

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  1. ^ Ares, Saul; Castro, Mario (February 1, 2006). "Hidden structure in the randomness of the prime number sequence?". Physica A: Statistical Mechanics and Its Applications. 360 (2): 285–296. arXiv:cond-mat/0310148. Bibcode:2006PhyA..360..285A. doi:10.1016/j.physa.2005.06.066. S2CID 16678116.
  2. ^ a b Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind", Commentationes Physico-Mathematicae Helsingsfors (in German), 5: 1–37, JFM 57.0186.02, Zbl 0003.24601.
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  5. ^ Andersen, Jens Kruse (March 8, 2013). "A megagap with merit 25.9". primerecords.dk. Archived from the original on December 25, 2019. Retrieved September 29, 2022.
  6. ^ a b c Nicely, Thomas R. (2019). "NEW PRIME GAP OF MAXIMUM KNOWN MERIT". faculty.lynchburg.edu. Archived from the original on April 30, 2021. Retrieved September 29, 2022.
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  20. ^ Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Yalçin (2010). "Primes in Tuples II". Acta Mathematica. 204 (1): 1–47. arXiv:0710.2728. doi:10.1007/s11511-010-0044-9. S2CID 7993099.
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  25. ^ Pintz, J. (1997). "Very large gaps between consecutive primes". J. Number Theory. 63 (2): 286–301. doi:10.1006/jnth.1997.2081.
  26. ^ Erdős, Paul; Bollobás, Béla; Thomason, Andrew, eds. (1997). Combinatorics, Geometry and Probability: A Tribute to Paul Erdős. Cambridge University Press. p. 1. ISBN 9780521584722. Archived from the original on September 29, 2022. Retrieved September 29, 2022.
  27. ^ Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence (2016). "Large gaps between consecutive prime numbers". Ann. of Math. 183 (3): 935–974. arXiv:1408.4505. doi:10.4007/annals.2016.183.3.4. MR 3488740. S2CID 16336889.
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  29. ^ Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2018). "Long gaps between primes". J. Amer. Math. Soc. 31 (1): 65–105. arXiv:1412.5029. doi:10.1090/jams/876. MR 3718451. S2CID 14487001.
  30. ^ Tao, Terence (December 16, 2014). "Long gaps between primes / What's new". Archived from the original on June 9, 2019. Retrieved August 29, 2019.
  31. ^ Ford, Kevin; Maynard, James; Tao, Terence (October 13, 2015). "Chains of large gaps between primes". arXiv:1511.04468 [math.NT].
  32. ^ a b Guy (2004) §A8
  33. ^ Cramér, Harald (1936). "On the order of magnitude of the difference between consecutive prime numbers". Acta Arithmetica. 2: 23–46. doi:10.4064/aa-2-1-23-46.
  34. ^ Ingham, Albert E. (1937). "On the difference between consecutive primes" (PDF). Quarterly Journal of Mathematics. 8 (1). Oxford: 255–266. Bibcode:1937QJMat...8..255I. doi:10.1093/qmath/os-8.1.255. Archived (PDF) from the original on December 5, 2022.
  35. ^ Sinha, Nilotpal Kanti (2010). "On a new property of primes that leads to a generalization of Cramer's conjecture". arXiv:1010.1399 [math.NT].
  36. ^ Kourbatov, Alexei (2015). "Upper bounds for prime gaps related to Firoozbakht's conjecture". Journal of Integer Sequences. 18 (11) 15.11.2. arXiv:1506.03042.
  37. ^ Granville, Andrew (1995). "Harald Cramér and the distribution of prime numbers" (PDF). Scandinavian Actuarial Journal. 1: 12–28. CiteSeerX 10.1.1.129.6847. doi:10.1080/03461238.1995.10413946. Archived (PDF) from the original on September 23, 2015. Retrieved March 2, 2016..
  38. ^ Granville, Andrew (1995). "Unexpected Irregularities in the Distribution of Prime Numbers" (PDF). Proceedings of the International Congress of Mathematicians. Vol. 1. pp. 388–399. doi:10.1007/978-3-0348-9078-6_32. ISBN 978-3-0348-9897-3. Archived (PDF) from the original on May 7, 2016. Retrieved March 2, 2016..
  39. ^ Pintz, János (September 2007). "Cramér vs. Cramér: On Cramér's probabilistic model for primes". Functiones et Approximatio Commentarii Mathematici. 37 (2): 232–471. doi:10.7169/facm/1229619660.

Further reading

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