“Landau's problems”的意思、由来-开放百科全书

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:

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.

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.

Twin prime conjecture

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

. A table of maximal prime gaps shows that the conjecture holds to 4×10¹⁸.^[11] 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

exceptional primes followed by gaps larger than

; in particular,

A result due to Ingham shows that there is a prime between

and

for every large enough n.^[13]

Near-square primes

Landau's fourth problem asked whether there are infinitely many primes which are of the form

for integer n. (The list of such primes is {{OEIS|id=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. {{As of|2018}}, 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.^[14]^[15] Nesmith Ankeny proved that, assuming the extended Riemann hypothesis for L-functions on Hecke characters, there are infinitely many primes of the form

with

.^[16] Landau's conjecture is for the stronger

Deshouillers and Iwaniec,^[17] improving on Hooley^[18] and Todd,^[19] 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 n² + 1 are composite.

Notes

1. ^{{cite arXiv |eprint=1305.2897 |title = Major arcs for Goldbach's theorem|last = Helfgott|first = H.A. |class=math.NT |year=2013}}
2. ^{{cite arXiv |eprint=1205.5252 |title = Minor arcs for Goldbach's problem |last = Helfgott|first = H.A.|class=math.NT |year=2012}}
3. ^{{cite arXiv |eprint=1312.7748 |title = The ternary Goldbach conjecture is true|last = Helfgott|first = H.A. |class=math.NT |year=2013}}
4. ^A semiprime is a natural number that is the product of two prime factors.
5. ^{{cite journal |first=H. L. |last=Montgomery |last2=Vaughan |first2=R. C. |url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27126.pdf |title=The exceptional set in Goldbach's problem |journal=Acta Arithmetica |volume=27 |issue= |year=1975 |pages=353–370 |doi= }}
6. ^{{cite arXiv|last=Yamada |first=Tomohiro |eprint=1511.03409 |title=Explicit Chen's theorem |class=math.NT |date=2015-11-11}}
7. ^Yitang Zhang, Bounded gaps between primes, Annals of Mathematics 179 (2014), pp. 1121–1174 from Volume 179 (2014), Issue 3
8. ^{{cite journal | author=D.H.J. Polymath | title=Variants of the Selberg sieve, and bounded intervals containing many primes | journal=Research in the Mathematical Sciences | volume=1 | number=12 | doi=10.1186/s40687-014-0012-7 | arxiv=1407.4897 | year=2014 | mr=3373710 | page=12}}
9. ^J. Maynard (2015), [https://arxiv.org/abs/1311.4600 Small gaps between primes]. Annals of Mathematics 181(1): 383-413.
10. ^{{cite journal | last1 = Alan Goldston | first1 = Daniel | last2 = Motohashi | first2 = Yoichi | last3 = Pintz | first3 = János | last4 = Yalçın Yıldırım | first4 = Cem | year = 2006 | title = Small Gaps between Primes Exist | url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pja/1146576181 | journal = Proceedings of the Japan Academy, Series A Mathematical Sciences | volume = 82 | issue = 4| pages = 61–65 | doi=10.3792/pjaa.82.61}}
11. ^Jens Kruse Andersen, Maximal Prime Gaps.
12. ^{{cite journal|author=Kaisa Matomäki|title=Large differences between consecutive primes|journal=Quarterly Journal of Mathematics|volume=58|year=2007|pages=489–518|doi=10.1093/qmath/ham021}}.
13. ^{{cite journal |first=A. E. |last=Ingham |title=On the difference between consecutive primes |journal=Quarterly Journal of Mathematics Oxford |volume=8 |year=1937 |issue=1 |pages=255–266 |doi=10.1093/qmath/os-8.1.255 |bibcode=1937QJMat...8..255I }}
14. ^{{cite journal |first=H. |last=Iwaniec|title=Almost-primes represented by quadratic polynomials|journal=Inventiones Mathematicae |volume=47 |issue=2 |year=1978|pages=178–188 |doi=10.1007/BF01578070 |bibcode=1978InMat..47..171I}}
15. ^{{cite journal|author=Robert J. Lemke Oliver|title=Almost-primes represented by quadratic polynomials|journal=Acta Arithmetica|volume=151|issue=3|year=2012|pages=241–261|url=http://www.stanford.edu/~rjlo/papers/04-Quadratic.pdf|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, [https://eudml.org/doc/74560 On the greatest prime factor of

], 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. ^{{citation|author=J. Todd|title=A problem on arc tangent relations|journal=American Mathematical Monthly |volume=56 |issue=8 |year=1949 |pages=517–528 |doi=10.2307/2305526 |jstor=2305526}}