Prime gap

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)-th 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 (gn) of prime gaps has been extensively studied.

The first 30 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 .

Simple observations

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

$P\#+2, P\#+3,\ldots,P\#+(Q-1)$

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 gnP. (This is seen by choosing n so that pn 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 twenty-seven 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 gn = 2 for infinitely many integers n.

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Numerical results

As of 2012 the largest known prime gap with identified probable prime gap ends has length 2254930, with 86853-digit probable primes found by H. Rosenthal and J. K. Andersen.[1] The largest known prime gap with identified proven primes as gap ends has length 337446, with 7996-digit primes found by T. Alm, J. K. Andersen and François Morain.[2]

We say that gn is a maximal gap if gm < gn for all m < n. As of August 2009 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.[3]

Usually the ratio of gn / ln(pn) is called the merit of the gap gn . As of January 2012, the largest known merit value is 66520 / ln(1931*1933#/7230 - 30244) ≈ 35.4244594 where 1933# indicates the primorial of 1933. This number, 1931*1933#/7230 - 30244, is a 816-digit prime. The next largest known merit value is 1476 / ln(1425172824437699411) ≈ 35.31.[4][5]

Prime gap function
The first 75 maximal gaps (n is not listed)
Number 1 to 25
# 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 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
Number 26 to 50
# gn pn
26 222 122164747
27 234 189695659
28 248 191912783
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
Number 51 to 75
# gn pn
51 582 1346294310749
52 588 1408695493609
53 602 1968188556461
54 652 2614941710599
55 674 7177162611713
56 716 13829048559701
57 766 19581334192423
58 778 42842283925351
59 804 90874329411493
60 806 171231342420521
61 906 218209405436543
62 916 1189459969825483
63 924 1686994940955803
64 1132 1693182318746371
65 1184 43841547845541059
66 1198 55350776431903243
67 1220 80873624627234849
68 1224 203986478517455989
69 1248 218034721194214273
70 1272 305405826521087869
71 1328 352521223451364323
72 1356 401429925999153707
73 1370 418032645936712127
74 1442 804212830686677669
75 1476 1425172824437699411
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Further results

Upper bounds

Bertrand's postulate 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 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 gn < εpn for all n > N.

One can deduce that the gaps get arbitrarily smaller in proportion to the primes: the quotient gn/pnapproaches zero as n goes to infinity.

Hoheisel was the first to show[6] 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

for sufficiently large n.

Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,[7] and to θ = 3/4 + ε, for any ε > 0, by Chudakov.[8]

A major improvement is due to Ingham,[9] 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 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. Note however that not even the Lindelöf hypothesis, which assumes that we can take c to be any positive number, implies that there is a prime number between n2 and (n + 1)2, if n is 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.[10]

A recent result, due to Baker, Harman and Pintz, shows that θ may be taken to be 0.525.[11]

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[12] to

$\liminf_{n\to\infty}\frac{g_n}{\sqrt{\log p_n}(\log\log p_n)^2}<\infty.$

On May 13, 2013, Yitang Zhang proposed a proof that $\liminf_{n\to\infty} g_n < 7\cdot 10^7$, meaning infinitely many gaps do not exceed 70 million.[13][14]

Lower bounds

Robert Rankin 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 with c = eγ. The best known value of the constant c is currently c = 2eγ, where γ is the Euler–Mascheroni constant.[15]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.[16]

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Even better results are possible if it is assumed that the Riemann hypothesis is true. Harald Cramér proved that, under this assumption, the gap g(pn) satisfies

$g(p_n) = O(\sqrt{p_n} \ln p_n),$

using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that

$g(p_n) = O\left((\ln p_n)^2\right).$

At the moment, the numerical evidence seems to point in this direction. See Cramér's conjecture for more details.

Andrica's conjecture states that

$g(p_n) < 2\sqrt{p_n} + 1.\,$

This is a slight strengthening of Legendre's conjecture that between successive square numbers there is always a prime.

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As an arithmetic function

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

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References

1. ^ Largest known prime gap
2. ^ A proven prime gap of 337446
3. ^ Maximal Prime Gaps
4. ^ The Top-20 Prime Gaps
5. ^ NEW PRIME GAP OF MAXIMUM KNOWN MERIT
6. ^ Hoheisel, G. (1930). "Primzahlprobleme in der Analysis". Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 33: 3–11.
7. ^ Heilbronn, H. A. (1933). "Über den Primzahlsatz von Herrn Hoheisel". Mathematische Zeitschrift 36 (1): 394–423. doi:10.1007/BF01188631.
8. ^ Tchudakoff, N. G. (1936). "On the difference between two neighboring prime numbers". Math. Sb. 1: 799–814.
9. ^ Ingham, A. E. (1937). "On the difference between consecutive primes". Quarterly Journal of Mathematics. Oxford Series 8 (1): 255–266. doi:10.1093/qmath/os-8.1.255.
10. ^ Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones Mathematicae 15 (2): 164–170. doi:10.1007/BF01418933.
11. ^ Baker, R. C.; Harman, G.; Pintz, 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.
12. ^
13. ^ McKee, Maggie (14 May 2013). "First proof that infinitely many prime numbers come in pairs". Nature. ISSN 0028-0836.
14. ^ Grossman, Lisa (14 May 2013). "Proof that an infinite number of primes are paired". New Scientist.
15. ^ Pintz, J. (1997). "Very large gaps between consecutive primes". J. Number Theory 63 (2): 286–301. doi:10.1006/jnth.1997.2081.
16. ^ a b Guy (2004) §A8
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