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, ${\displaystyle 2n=p+q}$ where p is prime and q is either prime or semiprime.^{[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 ${\displaystyle e^{e^{32.7}}\approx 1.4\cdot 10^{69057979807814}}$ is the sum of a prime and a product of at most two primes. Bordignon and Starichkova^[7] reduce this to ${\displaystyle e^{e^{15.85}}\approx 3.6\cdot 10^{3321634}}$ 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 395 primes rather than a prime or semiprime; under GRH they improve 395 to 31.^[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 ${\displaystyle E(x)<x^{0.72}}$ (for large enough x) due to Pintz,^[10][11] and ${\displaystyle E(x)\ll x^{0.5}\log ^{3}x}$ under RH, due to Goldston.^[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.^[13] Following many advances (see Pintz^[14] for an overview), Pintz and Ruzsa^[15] improved this to K = 8. Assuming the GRH, this can be improved to K = 7.^[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.^[20]

In 1966 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 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 ${\displaystyle x^{7/100+\varepsilon }}$ exceptional primes followed by gaps larger than ${\displaystyle {\sqrt {2p}}}$; in particular,

${\displaystyle \sum _{\stackrel {p_{n+1}-p_{n}>{\sqrt {p_{n}}}^{1/2}}{p_{n}\leq x}}p_{n+1}-p_{n}\ll x^{0.57+\varepsilon }.}$

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.^[25]

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 A002496.) 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.

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.^[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 ${\displaystyle p=x^{2}+y^{2}}$ with ${\displaystyle y=O(\log p)}$. Landau's conjecture is for the stronger ${\displaystyle y=1}$. The best unconditional result is due to Harman and Lewis^[30] and it gives ${\displaystyle y=O(p^{0.119})}$. In 2024, Ben Green and Mehtaab Sawhney proved that infinitely many primes are of the form ${\displaystyle p^{2}+nq^{2}}$ for any given ${\displaystyle n\equiv 0}$ or ${\displaystyle n\equiv 4}{\pmod {6}}$, noting that "the argument could surely be modified in a straightforward manner to handle arbitrary positive definite binary quadratic forms over ${\displaystyle \mathbb {Z} }$".^[31]

Grimmelt & Merikoski,^[32] improving on previous works,^{[33][34][35][36][37][38]} showed that there are infinitely many numbers of the form ${\displaystyle n^{2}+1}$ with greatest prime factor at least ${\displaystyle n^{1.312}}$. Replacing the exponent with 2 would yield Landau's conjecture.

The Friedlander–Iwaniec theorem shows that infinitely many primes are of the form ${\displaystyle x^{2}+y^{4}}$.^[39]

Baier and Zhao^[40] prove that there are infinitely many primes of the form ${\displaystyle p=an^{2}+1}$ with ${\displaystyle a<p^{5/9+\varepsilon }}$; the exponent can be improved to ${\displaystyle 1/2+\varepsilon }$ under the Generalized Riemann Hypothesis for L-functions and to ${\displaystyle \varepsilon }$ under a certain Elliott-Halberstam type hypothesis.

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 ${\displaystyle x}$. Hence almost all numbers of the form ${\displaystyle n^{2}+1}$ are composite.

• List of unsolved problems in mathematics
• Hilbert's problems