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.
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 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 A001223.
By the definition of gn the following sum can be stated as
The first, smallest, and only odd prime gap is 1 between the only even prime number, 2, and the first odd prime, 3. All other prime gaps are even.
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 gn ≥ P. (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.
As of 2013[update] the largest known prime gap with identified probable prime gap ends has length 3311852, with 97953-digit probable primes found by M. Jansen and J. K. Andersen. 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 gn is a maximal gap if gm < gn for all m < n. As of August 2009[update] 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 A002386.
Usually the ratio of gn / ln(pn) is called the merit of the gap gn . As of January 2012, the largest known merit value, as discovered by M. Jansen, 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. Other record merit terms can be found at A111870.
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
hence showing that
for sufficiently large n.
for some positive constant c, where O refers to the big O notation, then
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. 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.
and later improved it to
In 2013, Yitang Zhang proved that , 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. Their best reported bound has since been improved to 252 as of March 2014[update].
Robert Rankin proved the existence of a constant c > 0 such that the inequality
holds for infinitely many values n: he showed that one can take c = eγ, where γ is the Euler–Mascheroni constant. The best known value of the constant c is currently c = 2eγ. 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
using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that
At the moment, the numerical evidence seems to point in this direction. See Cramér's conjecture for more details.
The Firoozbakht’s conjecture which is a slight strengthening to Cramér's, satisfies
Mean while, the Oppermann's conjecture is a conjecture which is weaker than Cramér's conjecture. The expected gap size with Oppermann's conjecture is
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 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. The function is neither multiplicative nor additive.
- Largest known prime gap
- A proven prime gap of 1113106
- Maximal Prime Gaps
- The Top-20 Prime Gaps
- NEW PRIME GAP OF MAXIMUM KNOWN MERIT
- Hoheisel, G. (1930). "Primzahlprobleme in der Analysis". Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 33: 3–11. JFM 56.0172.02.
- Heilbronn, H. A. (1933). "Über den Primzahlsatz von Herrn Hoheisel". Mathematische Zeitschrift 36 (1): 394–423. doi:10.1007/BF01188631.
- Tchudakoff, N. G. (1936). "On the difference between two neighboring prime numbers". Math. Sb. 1: 799–814.
- 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.
- Cheng, Yuan-You Fu-Rui (2010). "Explicit estimate on primes between consecutive cubes". Rocky Mt. J. Math. 40: 117–153. Zbl 1201.11111.
- Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones Mathematicae 15 (2): 164–170. doi:10.1007/BF01418933.
- 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.
- "Primes in Tuples II". ArXiv. Retrieved 2013-11-23.
- Zhang, Yitang. "Bounded gaps between primes". Annals of Mathematics (Princeton University and the Institute for Advanced Study). Retrieved August 16, 2013.
- "Bounded gaps between primes". Polymath. Retrieved 2013-07-21.
- "Bounded gaps between primes". Polymath. Retrieved 2014-03-27.
- Pintz, J. (1997). "Very large gaps between consecutive primes". J. Number Theory 63 (2): 286–301. doi:10.1006/jnth.1997.2081.
- Guy (2004) §A8
- Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 978-0-387-20860-2. Zbl 1058.11001.
- 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.org.
- 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.