# Landau's problems

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 December 2019, all four problems are unresolved.

## Progress toward solutions

### Goldbach's conjecture

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.[5] The best current bound on the exceptional set is ${\displaystyle E(x)  (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.

### Twin prime conjecture

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.

### Legendre's conjecture

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]

### Near-square primes

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 such primes 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 n2 + 1 are composite.

## Notes

1. ^ Helfgott, H.A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
2. ^ Helfgott, H.A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
3. ^ Helfgott, H.A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
4. ^ A semiprime is a natural number that is the product of two prime factors.
5. ^ Montgomery, H. L.; Vaughan, R. C. (1975). "The exceptional set in Goldbach's problem" (PDF). Acta Arithmetica. 27: 353–370.
6. ^
7. ^ Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv:1511.03409 [math.NT].
8. ^ Yitang Zhang, Bounded gaps between primes, Annals of Mathematics 179 (2014), pp. 1121–1174 from Volume 179 (2014), Issue 3
9. ^ 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.
10. ^ J. Maynard (2015), Small gaps between primes. Annals of Mathematics 181(1): 383-413.
11. ^ Alan Goldston, Daniel; Motohashi, Yoichi; Pintz, János; Yalçın Yıldırım, Cem (2006). "Small Gaps between Primes Exist". Proceedings of the Japan Academy, Series A Mathematical Sciences. 82 (4): 61–65. arXiv:math/0505300. doi:10.3792/pjaa.82.61.
12. ^ Jens Kruse Andersen, Maximal Prime Gaps.
13. ^ Kaisa Matomäki (2007). "Large differences between consecutive primes". Quarterly Journal of Mathematics. 58: 489–518. doi:10.1093/qmath/ham021..
14. ^ 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.
15. ^ Iwaniec, H. (1978). "Almost-primes represented by quadratic polynomials". Inventiones Mathematicae. 47 (2): 178–188. Bibcode:1978InMat..47..171I. doi:10.1007/BF01578070.
16. ^ Robert J. Lemke Oliver (2012). "Almost-primes represented by quadratic polynomials" (PDF). Acta Arithmetica. 151 (3): 241–261. doi:10.4064/aa151-3-2..
17. ^ N. C. Ankeny, Representations of primes by quadratic forms, Amer. J. Math. 74:4 (1952), pp. 913–919.
18. ^ Jori Merikoski, Largest prime factor of n^2+1, 2019 preprint
19. ^ R. de la Bretèche and S. Drappeau. Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables. Journal of the European Mathematical Society, 2019.
20. ^ Jean-Marc Deshouillers and Henryk Iwaniec, On the greatest prime factor of ${\displaystyle n^{2}+1}$ , Annales de l'Institut Fourier 32:4 (1982), pp. 1–11.
21. ^ C. Hooley, On the greatest prime factor of a quadratic polynomial, Acta Math., 117 ( 196 7), 281–299.
22. ^ J. Todd (1949), "A problem on arc tangent relations", American Mathematical Monthly, 56 (8): 517–528, doi:10.2307/2305526, JSTOR 2305526
23. ^ J. Ivanov, Uber die Primteiler der Zahlen vonder Form A+x^2, Bull. Acad. Sci. St. Petersburg 3 (1895), 361–367.