Landau's problems

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers.

They are as follows: As of 2025[update], all four problems are unresolved.

Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldbach's conjecture.

Ivan Vinogradov proved it for large enough n (Vinogradov's theorem) in 1937,[1] and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013.

[2][3][4] Chen's theorem, another weakening of Goldbach's conjecture, proves that for all sufficiently large n,

[note 1] Bordignon, Johnston, and Starichkova,[5] correcting and improving on Yamada,[6] proved an explicit version of Chen's theorem: every even number greater than

assuming the Generalized Riemann hypothesis (GRH) for Dirichlet L-functions.

Johnston and Starichkova give a version working for all n ≥ 4 at the cost of using a number which is the product of at most 369 primes rather than a prime or semiprime; under GRH they improve 369 to 33.

[8] Montgomery and Vaughan showed that the exceptional set of even numbers not expressible as the sum of two primes has a density zero, although the set is not proven to be finite.

[9] The best current bounds on the exceptional set is

[12] Linnik proved that large enough even numbers could be expressed as the sum of two primes and some (ineffective) constant K of powers of 2.

[16] In 2013 Yitang Zhang showed[17] 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.

[18] Under the generalized Elliott–Halberstam conjecture this was improved to 6, extending earlier work by Maynard[19] and Goldston, Pintz and Yıldırım.

It suffices to check that each prime gap starting at p is smaller than

A table of maximal prime gaps shows that the conjecture holds to 264 ≈ 1.8×1019.

[21] A counterexample near that size would require a prime gap a hundred million times the size of the average gap.

Järviniemi,[22] improving on work by Heath-Brown[23] and by Matomäki,[24] shows that there are at most

exceptional primes followed by gaps larger than

; in particular, A result due to Ingham shows that there is a prime between

for every large enough n.[25] Landau's fourth problem asked whether there are infinitely many primes which are of the form

for integer n. (The list of known primes of this form is A002496.)

Henryk Iwaniec showed that there are infinitely many numbers of the form

[26][27] Ankeny[28] and Kubilius[29] proved that, assuming the extended Riemann hypothesis for L-functions on Hecke characters, there are infinitely many primes of the form

The best unconditional result is due to Harman and Lewis[30] and it gives

Merikoski,[31] improving on previous works,[32][33][34][35][36] showed that there are infinitely many numbers of the form

[note 2] Replacing the exponent with 2 would yield Landau's conjecture.

The Friedlander–Iwaniec theorem shows that infinitely many primes are of the form

[37] Baier and Zhao[38] prove that there are infinitely many primes of the form

under the Generalized Riemann Hypothesis for L-functions and to

The Brun sieve establishes an upper bound on the density of primes having the form