Prime gap

Bertrand's postulate, proven 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, proven in 1896, says that the average length of the gap between a prime p and the next prime will asymptotically approach ln(p) for sufficiently large primes. The actual length of the gap might be much more or less than this. However, one can deduce from the prime number theorem an upper bound on the length of prime gaps:

For every ${\displaystyle \epsilon >0}$, there is a number ${\displaystyle N}$ such that for all ${\displaystyle n>N}$

${\displaystyle g_{n}<p_{n}\epsilon }$.

One can also deduce that the gaps get arbitrarily smaller in proportion to the primes: the quotient

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

Hoheisel (1930) was the first to show[10] that there exists a constant θ < 1 such that

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

hence showing that

${\displaystyle g_{n}<p_{n}^{\theta },\,}$

for sufficiently large n.

Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,[11] and to θ = 3/4 + ε, for any ε > 0, by Chudakov.[12]

A major improvement is due to Ingham,[13] who showed that for some positive constant c, if

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

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.[14] 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).[15]

A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.[16]

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

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

and 2 years later improved this[17] to

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

In 2013, Yitang Zhang proved that

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

meaning that there are infinitely many gaps that do not exceed 70 million.[18] A Polymath Project collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013.[19] 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 with an infinite number of translations each of which containing m prime numbers.[20] Using Maynard's ideas, the Polymath project improved the bound to 246;[19][21] assuming the Elliott–Halberstam conjecture and its generalized form, N has been reduced to 12 and 6, respectively.[19]

In 1931, Erik Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,[2]

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

In 1938, Robert Rankin proved the existence of a constant c > 0 such that the inequality

${\displaystyle g_{n}>{\frac {c\log n\log \log n\log \log \log \log n}{(\log \log \log n)^{2}}}}$

holds for infinitely many values n, improving the results of 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.[22]

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.[23] This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.[24][25]

The result was further improved to

${\displaystyle g_{n}>{\frac {c\log n\log \log n\log \log \log \log n}{\log \log \log n}}}$

for infinitely many values of n by Ford–Green–Konyagin–Maynard–Tao.[26]

In the spirit of Erdös' original prize, Terence Tao offered 10,000 USD for a proof that c may be taken arbitrarily large in this inequality.[27]

Lower bounds for chains of primes have also been determined.[28]