Prime gap

Simple observations

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.

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

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 that 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 prime gap to the integers involved decreases (and is asymptotically zero). This is a consequence of the prime number theorem; see below. 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; see below.

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  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,

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  A111943.

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

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

hence showing that

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

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.

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

and 2 years 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 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

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

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

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

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

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.,

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  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 > 2 − ε e γ ( log ⁡ p n ) 2 infinitely often for any ε > 0 , where γ 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

As a result, there is (under Oppermann's conjecture) m>0 (probably m=30) for which every natural n>m satisfies g n < p n .

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

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.