Landau's problems

Vinogradov's theorem proves Goldbach's weak conjecture for sufficiently large n. In 2013, Harald Helfgott proved the weak conjecture for all odd numbers greater than 5.[1][2][3] Unlike Goldbach's conjecture, Goldbach's weak conjecture states that every odd number greater than 5 can be expressed as the sum of three primes. Although Goldbach's strong conjecture has not been proven or disproven, its proof would imply the proof of Goldbach's weak conjecture.

Chen's theorem proves that for all sufficiently large n, ${\displaystyle 2n=p+q}$ where p is prime and q is either prime or semiprime.[4] Montgomery and Vaughan showed that the exceptional set (even numbers not expressible as the sum of two primes) was of density zero, although the set is not proven to be finite.[5] The best current bound on the exceptional set is ${\displaystyle E(x)<x^{0.72}}$ (for large enough x) due to Pintz.[6]

In 2015, Tomohiro Yamada proved an explicit version of Chen's theorem:[7] every even number greater than ${\displaystyle e^{e^{36}}\approx 1.7\cdot 10^{1872344071119348}}$ is the sum of a prime and a product of at most two primes.

Yitang Zhang[8] showed that there are infinitely many prime pairs with gap bounded by 70 million, and this result has been improved to gaps of length 246 by a collaborative effort of the Polymath Project.[9] Under the generalized Elliott–Halberstam conjecture this was improved to 6, extending earlier work by Maynard[10] and Goldston, Pintz & Yıldırım.[11]

Chen showed that there are infinitely many primes p (later called Chen primes) such that p + 2 is either a prime or a semiprime.

It suffices to check that each prime gap starting at p is smaller than ${\displaystyle 2{\sqrt {p}}}$. A table of maximal prime gaps shows that the conjecture holds to 4 × 10¹⁸.[12] A counterexample near 10¹⁸ would require a prime gap fifty million times the size of the average gap. Matomäki shows that there are at most ${\displaystyle x^{1/6}}$ exceptional primes followed by gaps larger than ${\displaystyle {\sqrt {2p}}}$; in particular,

${\displaystyle \sum _{\stackrel {p_{n+1}-p_{n}>x^{1/2}}{x\leq p_{n}\leq 2x}}p_{n+1}-p_{n}\ll x^{2/3}.}$[13]

A result due to Ingham shows that there is a prime between ${\displaystyle n^{3}}$ and ${\displaystyle (n+1)^{3}}$ for every large enough n.[14]

Landau's fourth problem asked whether there are infinitely many primes which are of the form ${\displaystyle p=n^{2}+1}$ for integer n. (The list of known primes of this form is (sequence A002496 in the OEIS).) The existence of infinitely many such primes would follow as a consequence of other number-theoretic conjectures such as the Bunyakovsky conjecture and Bateman–Horn conjecture. As of 2020, this problem is open.

One example of near-square primes are Fermat primes. Henryk Iwaniec showed that there are infinitely many numbers of the form ${\displaystyle n^{2}+1}$ with at most two prime factors.[15][16] Nesmith Ankeny proved that, assuming the extended Riemann hypothesis for L-functions on Hecke characters, there are infinitely many primes of the form ${\displaystyle x^{2}+y^{2}}$ with ${\displaystyle y=O(\log x)}$.[17] Landau's conjecture is for the stronger ${\displaystyle y=1}$.

Merikoski,[18] improving on previous works,[19][20][21][22][23] showed that there are infinitely many numbers of the form ${\displaystyle n^{2}+1}$ with greatest prime factor at least ${\displaystyle n^{1.279}}$. Replacing the exponent with 2 would yield Landau's conjecture.

The Brun sieve establishes an upper bound on the density of primes having the form ${\displaystyle p=n^{2}+1}$: there are ${\displaystyle O({\sqrt {x}}/\log x)}$ such primes up to ${\displaystyle x}$. It then follows that almost all numbers of the form ${\displaystyle n^{2}+1}$ are composite.