Prime gap

Prime gap frequency distribution for primes up to 1.6 billion. Peaks occur at multiples of 6.

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, 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 g_n every prime can be written as

$p_{n+1}=2+\sum _{i=1}^{n}g_{i}.$

Simple observations

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

$n!+2,n!+3,\ldots ,n!+n$

the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of n − 1 consecutive composite integers, and it must belong to a gap between primes having length at least n − 1. It follows that there are gaps between primes that are arbitrarily large, that is, for any integer N, there is an integer m with g_m ≥ N.

In reality, 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.

Although the average gap between primes increases as the natural logarithm of the integer, the ratio of the prime gap to the integers involved decreases (and is asymptotically zero). This is a consequence of the prime number theorem. On the other hand, 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.

In the opposite direction, the twin prime conjecture asserts that g_n = 2 for infinitely many integers n.

Numerical results

As of March 2017 the largest known prime gap with identified probable prime gap ends has length 5103138, with 216849-digit probable primes found by Robert W. Smith. This gap has merit M=10.2203. The largest known prime gap with identified proven primes as gap ends has length 1113106, with 18662-digit primes found by P. Cami, M. Jansen and J. K. Andersen.

We say that g_n is a maximal gap, if g_m < g_n for all m < n. As of August 2016 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. Other record maximal gap terms can be found at OEIS: A002386.

Usually the ratio of g_n / ln(p_n) is called the merit of the gap g_n . In 1931, E. Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,

$\limsup _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=\infty .$

Largest known merit values (As of November 2016)

Merit	gn	digits	pn	Date	Discoverer
36.858288	10716	127	7910896513*283#/30 - 6480	2016	Dana Jacobsen
36.590183	13692	163	1037600971*383#/210 - 8776	2016	Dana Jacobsen
36.420568	26892	321	59740589*757#/210 - 14302	2016	Dana Jacobsen
35.424459	66520	816	1931*1933#/7230 - 30244	2012	Michiel Jansen
35.310308	1476	19	1425172824437699411	2009	Tomás Oliveira e Silva

As of November 2016, the largest known merit value, as discovered by D. Jacobsen, is 10716 / ln(7910896513*283#/30 - 6480) ≈ 36.858288 where 283# indicates the primorial of 283. The endpoints are 127-digit primes.

The Cramér–Shanks–Granville ratio is the ratio of g_n / (ln(p_n))^2. The greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. Other record terms can be found at OEIS: A111943.

The first 75 maximal gaps

Number 1 to 25 # gn pn n 1 1 2 1 2 2 3 2 3 4 7 4 4 6 23 9 5 8 89 24 6 14 113 30 7 18 523 99 8 20 887 154 9 22 1,129 189 10 34 1,327 217 11 36 9,551 1,183 12 44 15,683 1,831 13 52 19,609 2,225 14 72 31,397 3,385 15 86 155,921 14,357 16 96 360,653 30,802 17 112 370,261 31,545 18 114 492,113 40,933 19 118 1,349,533 103,520 20 132 1,357,201 104,071 21 148 2,010,733 149,689 22 154 4,652,353 325,852 23 180 17,051,707 1,094,421 24 210 20,831,323 1,319,945 25 220 47,326,693 2,850,174

Number 26 to 50 # gn pn n 26 222 122,164,747 6,957,876 27 234 189,695,659 10,539,432 28 248 191,912,783 10,655,462 29 250 387,096,133 20,684,332 30 282 436,273,009 32,162,398 31 288 1,294,268,491 64,955,634 32 292 1,453,168,141 72,507,380 33 320 2,300,942,549 112,228,683 34 336 3,842,610,773 182,837,804 35 354 4,302,407,359 203,615,628 36 382 10,726,904,659 486,570,087 37 384 20,678,048,297 910,774,004 38 394 22,367,084,959 981,765,347 39 456 25,056,082,087 1,094,330,259 40 464 42,652,618,343 1,820,471,368 41 468 127,976,334,671 5,217,031,687 42 474 182,226,896,239 7,322,882,472 43 486 241,160,624,143 9,583,057,667 44 490 297,501,075,799 11,723,859,927 45 500 303,371,455,241 11,945,986,786 46 514 304,599,508,537 11,992,433,550 47 516 416,608,695,821 16,202,238,656 48 532 461,690,510,011 17,883,926,781 49 534 614,487,453,523 23,541,455,083 50 540 738,832,927,927 28,106,444,830

Number 51 to 75 # gn pn n 51 582 1,346,294,310,749 50,070,452,577 52 588 1,408,695,493,609 52,302,956,123 53 602 1,968,188,556,461 72,178,455,400 54 652 2,614,941,710,599 94,906,079,600 55 674 7,177,162,611,713 251,265,078,335 56 716 13,829,048,559,701 473,258,870,471 57 766 19,581,334,192,423 662,221,289,043 58 778 42,842,283,925,351 1,411,461,642,343 59 804 90,874,329,411,493 2,921,439,731,020 60 806 171,231,342,420,521 5,394,763,455,325 61 906 218,209,405,436,543 6,822,667,965,940 62 916 1,189,459,969,825,483 35,315,870,460,455 63 924 1,686,994,940,955,803 49,573,167,413,483 64 1,132 1,693,182,318,746,371 49,749,629,143,526 65 1,184 43,841,547,845,541,059 1,175,662,926,421,598 66 1,198 55,350,776,431,903,243 1,475,067,052,906,945 67 1,220 80,873,624,627,234,849 2,133,658,100,875,638 68 1,224 203,986,478,517,455,989 5,253,374,014,230,870 69 1,248 218,034,721,194,214,273 5,605,544,222,945,291 70 1,272 305,405,826,521,087,869 7,784,313,111,002,702 71 1,328 352,521,223,451,364,323 8,952,449,214,971,382 72 1,356 401,429,925,999,153,707 10,160,960,128,667,332 73 1,370 418,032,645,936,712,127 10,570,355,884,548,334 74 1,442 804,212,830,686,677,669 20,004,097,201,301,079 75 1,476 1,425,172,824,437,699,411 34,952,141,021,660,495

Further results

Upper bounds

Bertrand's postulate, proved 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, proved in 1896, 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

$\lim _{n\to \infty }{\frac {g_{n}}{p_{n}}}=0.$

Hoheisel (1930) was the first to show that there exists a constant θ < 1 such that

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

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 for some positive constant c, if

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

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. 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).

A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.

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

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

and 2 years later improved this to

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

In 2013, Yitang Zhang proved that

$\liminf _{n\to \infty }g_{n}<7\cdot 10^{7},$

meaning that there are infinitely many gaps that do not exceed 70 million. A Polymath Project collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013. 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 containing m prime numbers. Using Maynard's ideas, the Polymath project improved the bound to 246; assuming the Elliott–Halberstam conjecture and its generalized form, N has been reduced to 12 and 6, respectively.

Lower bounds

In 1938, 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, improving results by Erik 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.

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. This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.

The result was further improved to

$g_{n}\gg {\frac {\log n\log \log n\log \log \log \log n}{\log \log \log n}}$

for infinitely many values of n by Ford–Green–Konyagin–Maynard–Tao.

Lower bounds for chains of primes have also been determined.

Conjectures about gaps between primes

Prime gap function

Even better results are possible under the Riemann hypothesis. Harald Cramér proved that the Riemann hypothesis implies the gap g_n satisfies

$g_{n}=O({\sqrt {p_{n}}}\log p_{n}),$

using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking, Cramér's conjecture states that

$g_{n}=O\left((\log p_{n})^{2}\right).$

Firoozbakht's conjecture states that $p_{n}^{1/n}$ (where $p_{n}$ is the nth prime) is a strictly decreasing function of n, i.e.,

$p_{n+1}^{1/(n+1)}<p_{n}^{1/n}{\text{ for all }}n\geq 1.$

If this conjecture is true, then the function $g_{n}=p_{n+1}-p_{n}$ satisfies $g_{n}<(\log p_{n})^{2}-\log p_{n}{\text{ for all }}n>4.$ It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz which suggest that $g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}$ infinitely often for any $\varepsilon >0,$ where $\gamma$ denotes the Euler–Mascheroni constant.

Meanwhile, Oppermann's conjecture is weaker than Cramér's conjecture. The expected gap size with Oppermann's conjecture is on the order of

$g_{n}<{\sqrt {p_{n}}}.$

As a result, there is (under Oppermann's conjecture) m>0 (probably m=30) for which every natural n>m satisfies $g_{n}<{\sqrt {p_{n}}}.$

Andrica's conjecture, which is a weaker conjecture than Oppermann's, states that

$g_{n}<2{\sqrt {p_{n}}}+1.$

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

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 Zhang Yitang result proves that it is true for at least one (currently unknown) value of k which is smaller than 70,000,000.

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
• Twin prime

References

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

Further reading

• Soundararajan, Kannan (2007). "Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım". Bull. Am. Math. Soc., New Ser. 44 (1): 1–18. doi:10.1090/s0273-0979-06-01142-6. Zbl 1193.11086.
• Mihăilescu, Preda (June 2014). "On some conjectures in additive number theory" (PDF). Newsletter of the European Mathematical Society (92): 13–16. doi:10.4171/NEWS. ISSN 1027-488X.

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, does not involve an 'arbitrarily big' constant as some other reported results.
• Chris Caldwell, Gaps Between Primes; an elementary introduction
• www.primegaps.com A study of the gaps between consecutive prime numbers
• Andrew Granville, Primes in Intervals of Bounded Length; overview of the results obtained so far up to and including James Maynard's work of November 2013.