Prime gap

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A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted g_n, is the difference between the (n+1)-th and the n-th prime number, 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. One also writes g(p_n) for g_n.

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 (sequence A001223 in OEIS)

1 Simple observations
2 Numerical results
3 Further results
4 Conjectures about gaps between primes
5 References
6 External link

[edit] Simple observations

For any prime number P larger than 2, the sequence (for the notation P# see primorial, and P_n+1 denotes the smallest prime number greater than P)

P# + 2, P# + 3, ..., P# + (P_n+1-1)

is a sequence of P_n+1-2 consecutive composite integers, implying a prime gap of at least length P_n+1-1. Therefore, there exist gaps between primes which are arbitrarily large, i.e. for any prime number P, there is an integer n with g_p > P (This is seen by choosing n so that p_n is the greatest prime number less than P# + 2).

In reality, prime gaps of n numbers can occur at numbers much smaller than n#. For instance, the smallest sequence of 71 consecutive composite numbers occurs between 31398 and 31468, wheareas the number 71 primorial (the product of all prime numbers up to and including 71, or 71*67*61*59*53*47...*5*3*2) 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 g_n = 2 for infinitely many integers n.

[edit] Numerical results

As of 2007 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 g_n is a maximal gap if g_m < g_n for all m < n. As of December 2006 the largest known maximal gap has length 1370, found by Donald E. Knuth. It is the 73rd maximal gap, and it occurs after the prime 418032645936712127. [3]

The first 73 maximal gaps (n is not listed)

Number 1 to 25
#	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

Number 26 to 50
#	g_n	p_n
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 73
#	g_n	p_n
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
Gap with unknown status:
?	1442	804212830686677669

The largest known value of g_n / log(p_n) -- usually called the merit of the gap g_n -- is 1442 / log(804212830686677669) = 34.98, found by S. Herzog and T. O. e Silva. [4] As of December 2006 it is not known whether this is a maximal gap.

[edit] Further results

It follows from Bertrand's postulate that 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. Of course, the actual length of the gap might be much more or less than this. However, from the prime number theorem one can also easily 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.

Hoheisel was the first to show^[1] that there exists a constant θ < 1 such that

π(x + x^θ) - π(x) ~ x^θ/log(x), as x tends to infinity,

hence showing that

$g_n<p_n^\theta,$

if n is sufficiently large.

One deduces that the gaps get arbitrarily small in proportion to the primes: the quotient g_n/p_n approaches zero as n goes to infinity.

Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,^[2] and to θ = 3/4 + e, for any e > 0, by Čudakov.^[3]

A major improvement is due to Ingham,^[4] who showed that if

ζ(1/2 + it) = O(t^c),

for some positive constant c, then

π(x + x^θ) - π(x) ~ x^θ/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 (Compare Legendre's conjecture). To verify this, as of 2006 still unresolved problem, a stronger result such as Cramér's conjecture would be needed.

Huxley showed that one may choose θ = 7/12.^[5]

A recent result, due to Baker, Harman and Pintz, shows that θ may be taken to be 0.525.^[6]

[edit] 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 g(p) satisfies

$g(p) = O(\sqrt{p} \ln p).$

Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that

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

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

Andrica's conjecture states that

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

[edit] References

^ G. Hoheisel, Primzahlprobleme in der Analysis, Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 33, pages 3-11, (1930)
^ H. A. Heilbronn, Über den Primzahlsatz von Herrn Hoheisel, Mathematische Zeitschrift, 36, pages 394-423, (1933)
^ N. G. Tchudakoff, On the difference between two neighboring prime numbers, Math. Sb., 1, pages 799-814, (1936)
^ Ingham, A. E. On the difference between consecutive primes, Quarterly Journal of Mathematics (Oxford Series), 8, pages 255-266, (1937)
^ Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones mathematicae 15: 164-170.
^ Baker, R. C.; G. Harman, G. and J. Pintz (2001). "The difference between consecutive primes, II". Proceedings of the London Mathematical Society 83: 532–562.