At the 1912International Congress of Mathematicians,Edmund Landau listed four basic problems aboutprime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known asLandau's problems. They are as follows:
Chen's theorem, another weakening of Goldbach's conjecture, proves that for all sufficiently largen, wherep is prime andq is either prime orsemiprime.[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 is the sum of a prime and a product of at most two primes. Bordignon and Starichkova[7] reduce this to assuming theGeneralized Riemann hypothesis (GRH) forDirichlet L-functions. Johnston and Starichkova give a version working for alln ≥ 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 andVaughan showed that the exceptional set of even numbers not expressible as the sum of two primes has adensity zero, although the set is not proven to be finite.[9] The best current bounds on the exceptional set is (for large enoughx) due toPintz,[10][11] and underRH, due toGoldston.[12]
Linnik proved that large enough even numbers could be expressed as the sum of two primes and some (ineffective) constantK of powers of 2.[13] Following many advances (see Pintz[14] for an overview),Pintz andRuzsa[15] improved this toK = 8. Assuming the GRH, this can be improved toK = 7.[16]
It suffices to check that each prime gap starting atp is smaller than. A table of maximal prime gaps shows that theconjecture holds to 264 ≈ 1.8×1019.[21] Acounterexample 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 toIngham shows that there is a prime between and for every large enoughn.[25]
Landau's fourth problem asked whether there are infinitely many primes which are of the form for integern. (The list of known primes of this form isA002496.) The existence of infinitely many such primes would follow as a consequence of other number-theoretic conjectures such as theBunyakovsky conjecture andBateman–Horn conjecture.
One example of near-square primes areFermat primes.Henryk Iwaniec showed that there are infinitely many numbers of the form with at most two prime factors.[26][27]Ankeny[28] andKubilius[29] proved that, assuming theextended Riemann hypothesis forL-functions onHecke characters, there are infinitely many primes of the form with. Landau's conjecture is for the stronger. The best unconditional result is due to Harman and Lewis[30] and it gives. In 2024,Ben Green and Mehtaab Sawhney proved that infinitely many primes are of the form for any given or, noting that "the argument could surely be modified in a straightforward manner to handle arbitrary positive definitebinary quadratic forms over".[31]
Grimmelt & Merikoski,[32] improving on previous works,[33][34][35][36][37][38] showed that there are infinitely many numbers of the form with greatest prime factor at least. Replacing the exponent with 2 would yield Landau's conjecture.
Baier and Zhao[40] prove that there are infinitely many primes of the form with; the exponent can be improved to under the Generalized Riemann Hypothesis for L-functions and to under a certainElliott-Halberstam type hypothesis.
TheBrun sieve establishes an upper bound on the density of primes having the form: there are such primes up to. Hencealmost all numbers of the form are composite.
^Pintz, Janos (2018). "A new explicit formula in the additive theory of primes with applications II. The exceptional set in Goldbach's problem".arXiv:1804.09084 [math.NT].
^Goldston, D.A. (1992).On Hardy and Littlewood's contribution to the Goldbach conjecture. Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989). Università di Salerno. pp. 115–155.
^Yu V Linnik, Prime numbers and powers of two,Trudy Matematicheskogo Instituta imeni VA Steklova38 (1951), pp. 152-169.
^János Pintz, Approximations to the Goldbach and twin prime problem and gaps between consecutive primes, Probability and Number Theory (Kanazawa, 2005), Advanced Studies in Pure Mathematics 49, pp. 323–365. Math. Soc. Japan, Tokyo, 2007.
^Matomaki, K. (2007). "Large differences between consecutive primes".The Quarterly Journal of Mathematics.58 (4):489–518.doi:10.1093/qmath/ham021.ISSN0033-5606..
^Green, Ben; Sawhney, Mehtaab (2024). "Primes of the form $p^2 + nq^2$".arXiv:2410.04189 [math.NT].
^Grimmelt, Lasse; Merikoski, Jori (2025). "On the greatest prime factor and uniform equidistribution of quadratic polynomials".arXiv:2505.00493 [math.NT].
