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. There is only one pair of gaps between three consecutive odd natural numbers for which all are prime. These gaps are g2 and g3 between the primes 3, 5, and 7.
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 2014[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 June 2014[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 an 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. 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 there exists a bounded interval containing prime numbers. Using Maynard's ideas, the Polymath project has since improved the bound to 252 as of March 2014[update]. As of 14 April 2014[update], one year after Zhang's announcement, according to the Polymath project wiki, N has been reduced to 246. Further, assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that N has been reduced to 12 and 6, respectively.
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. This was proved independently by Ford-Green-Konyagin-Tao and James Maynard, in the positive, by two papers respectively sent to arXiv in 2014.
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.
Firoozbakht's conjecture states that (where is the nth prime) is a strictly decreasing function of n, i.e.,
If this conjecture is true, then the function satisfies 
This is the strongest upper bound ever conjectured for prime gaps. Moreover, this conjecture implies Cramér's conjecture in a strong form and would be consistent with Daniel Shanks conjectured asymptotic equality of record gaps.
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.
- Andersen, Jens Kruse. "The Top-20 Prime Gaps". Retrieved 2014-06-13.
- Largest known prime gap
- A proven prime gap of 1113106
- Maximal Prime Gaps
- NEW PRIME GAP OF MAXIMUM KNOWN MERIT
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