Prime gap

A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted g_n or g(p_n) 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 (g_n) 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 OEIS: A001223.

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 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 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.

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. 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.

We say that g_n is a maximal gap if g_m < g_n 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.

Usually the ratio of g_n / ln(p_n) is called the merit of the gap g_n . 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.

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

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 g_n/p_n approaches zero as n goes to infinity.

Hoheisel was the first to show 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, and to θ = 3/4 + ε, for any ε > 0, by Chudakov.

A major improvement is due to Ingham, 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.

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

In 2005, Daniel Goldston, János Pintz and Cem Yıldırım have proved that

$\liminf _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0$

and later improved it to

$\liminf _{n\to \infty }{\frac {g_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty$

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 = e^γ. The best known value of the constant c is currently c = 2e^γ, where γ is the Euler–Mascheroni constant. 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.

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_n) 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.

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. The function is neither multiplicative nor additive.

See also

• Bonse's inequality
• Gaussian moat

References

• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 978-0-387-20860-2. Zbl 1058.11001.

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". PlanetMath.
• Armin Shams, [Re-extending Chebyshev's theorem about Bertrand's conjecture http://link.springer.com/article/10.1007%2Fs11253-008-0034-7], does not involve an 'arbitrarily big' constant as some other reported results.
• Chris Caldwell, Gaps Between Primes
• www.primegaps.com A study of the gaps between consecutive prime numbers