# Landau's problems

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes. 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? (sequence A002496 in the OEIS).

As of 2017, 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. Montgomery and Vaughan showed that the exceptional set (even numbers not expressible as the sum of two primes) was of density zero.[4]

Tomohiro Yamada proved an explicit version of Chen's theorem:[5] 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[6] 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.[7] Under the generalized Elliott–Halberstam conjecture this was improved to 6, extending earlier work by Maynard[8] and Goldston, Pintz & Yıldırım.[9]

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.[10] 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,

${\displaystyle \sum _{\stackrel {p_{n+1}-p_{n}>x^{1/2}}{x\leq p_{n}\leq 2x}}p_{n+1}-p_{n}\ll x^{2/3}.}$[11]

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.[12]

### Near-square primes

The Friedlander–Iwaniec theorem shows that infinitely many primes are of the form ${\displaystyle x^{2}+y^{4}}$.[13]

Iwaniec showed that there are infinitely many numbers of the form ${\displaystyle n^{2}+1}$ with at most two prime factors.[14][15]

Ankeny proved that, under the extended Riemann hypothesis for L-functions on Hecke characters, there are infinitely many primes of the form ${\displaystyle x^{2}+y^{2}}$ with ${\displaystyle y=O(\log x)}$.[16] The conjecture is the stronger ${\displaystyle y=1}$.

Deshouillers & Iwaniec,[17] improving on Hooley[18] and Todd,[19] show that there are infinitely many numbers of the form ${\displaystyle n^{2}+1}$ with greatest prime factor at least ${\displaystyle n^{1.2}}$. Replacing the exponent with 2 would yield the conjecture.

In the opposite direction, the Brun sieve shows that there are ${\displaystyle O({\sqrt {x}}/\log x)}$ such primes up to x.

## Notes

1. ^ Helfgott, H.A. (2013). "Major arcs for Goldbach's theorem". arXiv: [math.NT].
2. ^ Helfgott, H.A. (2012). "Minor arcs for Goldbach's problem". arXiv: [math.NT].
3. ^ Helfgott, H.A. (2013). "The ternary Goldbach conjecture is true". arXiv: [math.NT].
4. ^ Montgomery, H. L.; Vaughan, R. C. (1975). "The exceptional set in Goldbach's problem" (PDF). Acta Arithmetica. 27: 353–370.
5. ^ Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv: [math.NT].
6. ^ Yitang Zhang, Bounded gaps between primes, Annals of Mathematics 179 (2014), pp. 1121–1174 from Volume 179 (2014), Issue 3
7. ^ 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:. doi:10.1186/s40687-014-0012-7. MR 3373710.
8. ^ J. Maynard, Small gaps between primes. To appear, Annals of Mathematics.
9. ^ 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. doi:10.3792/pjaa.82.61.
10. ^ Jens Kruse Andersen, Maximal Prime Gaps.
11. ^ Kaisa Matomäki (2007). "Large differences between consecutive primes". Quarterly Journal of Mathematics. 58: 489–518. doi:10.1093/qmath/ham021..
12. ^ Ingham, A. E. (1937). "On the difference between consecutive primes". Quarterly Journal of Mathematics Oxford. 8 (1): 255–266. doi:10.1093/qmath/os-8.1.255.
13. ^ Friedlander, John; Iwaniec, Henryk (1997), "Using a parity-sensitive sieve to count prime values of a  polynomial", PNAS, 94 (4): 1054–1058, doi:10.1073/pnas.94.4.1054, PMC , PMID 11038598.
14. ^ Iwaniec, H. (1978). "Almost-primes represented by quadratic polynomials". Inventiones Mathematicae. 47 (2): 178–188. doi:10.1007/BF01578070.
15. ^ Robert J. Lemke Oliver (2012). "Almost-primes represented by quadratic polynomials" (PDF). Acta Arithmetica. 151: 241–261. doi:10.4064/aa151-3-2..
16. ^ N. C. Ankeny, Representations of primes by quadratic forms, Amer. J. Math. 74:4 (1952), pp. 913–919.
17. ^ 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.
18. ^ C. Hooley, On the greatest prime factor of a quadratic polynomial, Acta Math., 117 ( 196 7), 281-299.
19. ^ J. Todd (1949), "A problem on arc tangent relations", American Mathematical Monthly, 56: 517–528, doi:10.2307/2305526