Landau's problems

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At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:

  1. Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
  2. Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
  3. Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
  4. Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n2 + 1?

As of August 2021, all four problems are unresolved.

Progress toward solutions[edit]

Goldbach's conjecture[edit]

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, where p is prime and q is either prime or semiprime.[5] In 2015, Tomohiro Yamada proved an explicit version of Chen's theorem:[6] every even number greater than is the sum of a prime and a product of at most two primes.

Montgomery and Vaughan showed that the exceptional set of even numbers not expressible as the sum of two primes was of density zero, although the set is not proven to be finite.[7] The best current bound on the exceptional set is (for large enough x) due to Pintz.[8]

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.[9] Following many advances (see Pintz[10] for an overview), Pintz and Ruzsa[11] improved this to K = 8.

Twin prime conjecture[edit]

Yitang Zhang[12] 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.[13] Under the generalized Elliott–Halberstam conjecture this was improved to 6, extending earlier work by Maynard[14] and Goldston, Pintz & Yıldırım.[15]

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

Legendre's conjecture[edit]

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

Matomäki shows that there are at most exceptional primes followed by gaps larger than ; in particular,

[17]

A result due to Ingham shows that there is a prime between and for every large enough n.[18]

Near-square primes[edit]

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 (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 with at most two prime factors.[19][20] Nesmith Ankeny proved that, assuming the extended Riemann hypothesis for L-functions on Hecke characters, there are infinitely many primes of the form with .[21] Landau's conjecture is for the stronger .

Merikoski,[22] improving on previous works,[23][24][25][26][27] 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.

The Brun sieve establishes an upper bound on the density of primes having the form : there are such primes up to . It then follows that almost all numbers of the form are composite.

See also[edit]

Notes[edit]

  1. ^ I. M. Vinogradov. Representation of an odd number as a sum of three primes, Doklady Akademii Nauk SSSR, 15 (1937), pp. 291-294.
  2. ^ Helfgott, H.A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
  3. ^ Helfgott, H.A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
  4. ^ Helfgott, H.A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
  5. ^ A semiprime is a natural number that is the product of two prime factors.
  6. ^ Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv:1511.03409 [math.NT].
  7. ^ Montgomery, H. L.; Vaughan, R. C. (1975). "The exceptional set in Goldbach's problem" (PDF). Acta Arithmetica. 27: 353–370. doi:10.4064/aa-27-1-353-370.
  8. ^ Janos Pintz, A new explicit formula in the additive theory of primes with applications II. The exceptional set in Goldbach's problem, 2018 preprint
  9. ^ Yu V Linnik, Prime numbers and powers of two, Trudy Matematicheskogo Instituta imeni VA Steklova 38 (1951), pp. 152-169.
  10. ^ 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.
  11. ^ Pintz, J.; Ruzsa, I. Z. (July 2020). "On Linnik's approximation to Goldbach's problem. II". Acta Mathematica Hungarica. 161: 569–582. doi:10.1007/s10474-020-01077-8.
  12. ^ Yitang Zhang, Bounded gaps between primes, Annals of Mathematics 179 (2014), pp. 1121–1174 from Volume 179 (2014), Issue 3
  13. ^ D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences. 1 (12): 12. arXiv:1407.4897. doi:10.1186/s40687-014-0012-7. MR 3373710. S2CID 119699189.
  14. ^ J. Maynard (2015), Small gaps between primes. Annals of Mathematics 181(1): 383-413.
  15. ^ Alan Goldston, Daniel; Motohashi, Yoichi; Pintz, János; Yalçın Yıldırım, Cem (2006). "Small Gaps between Primes Exist". Proceedings of the Japan Academy, Series A. 82 (4): 61–65. arXiv:math/0505300. doi:10.3792/pjaa.82.61. S2CID 18847478.
  16. ^ Dr. Thomas R. Nicely, First occurrence prime gaps
  17. ^ Kaisa Matomäki (2007). "Large differences between consecutive primes". Quarterly Journal of Mathematics. 58 (4): 489–518. doi:10.1093/qmath/ham021..
  18. ^ Ingham, A. E. (1937). "On the difference between consecutive primes". Quarterly Journal of Mathematics Oxford. 8 (1): 255–266. Bibcode:1937QJMat...8..255I. doi:10.1093/qmath/os-8.1.255.
  19. ^ Iwaniec, H. (1978). "Almost-primes represented by quadratic polynomials". Inventiones Mathematicae. 47 (2): 178–188. Bibcode:1978InMat..47..171I. doi:10.1007/BF01578070. S2CID 122656097.
  20. ^ Robert J. Lemke Oliver (2012). "Almost-primes represented by quadratic polynomials" (PDF). Acta Arithmetica. 151 (3): 241–261. doi:10.4064/aa151-3-2..
  21. ^ N. C. Ankeny, Representations of primes by quadratic forms, Amer. J. Math. 74:4 (1952), pp. 913–919.
  22. ^ Jori Merikoski, Largest prime factor of n^2+1, 2019 preprint
  23. ^ de la Bretèche, Régis; Drappeau, Sary (2020), "Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables", Journal of the European Mathematical Society, 22 (5): 1577–1624, arXiv:1703.03197, doi:10.4171/JEMS/951
  24. ^ Jean-Marc Deshouillers and Henryk Iwaniec, On the greatest prime factor of , Annales de l'Institut Fourier 32:4 (1982), pp. 1–11.
  25. ^ Hooley, Christopher (July 1967). "On the greatest prime factor of a quadratic polynomial". Acta Mathematica. 117: 281–299. doi:10.1007/BF02395047.
  26. ^ J. Todd (1949), "A problem on arc tangent relations", American Mathematical Monthly, 56 (8): 517–528, doi:10.2307/2305526, JSTOR 2305526
  27. ^ J. Ivanov, Uber die Primteiler der Zahlen vonder Form A+x^2, Bull. Acad. Sci. St. Petersburg 3 (1895), 361–367.

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