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| At the 1912 [[International Congress of Mathematicians]], [[Edmund Landau]] listed four basic problems about [[prime number|primes]]. These problems were characterised in his speech as "unattackable at the present state of science" and are now known as '''Landau's problems'''. They are as follows:
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| # [[Goldbach's conjecture]]: Can every even integer greater than 2 be written as the sum of two primes?
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| # [[Twin prime conjecture]]: Are there infinitely many primes ''p'' such that ''p'' + 2 is prime?
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| # [[Legendre's conjecture]]: Does there always exist at least one prime between consecutive [[Square number|perfect squares]]?
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| # 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 ''n''<sup>2</sup> + 1? {{OEIS|id=A002496}}.
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| {{As of|2013}}, all four problems are unresolved.
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| ==Progress toward solutions==
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| ===Goldbach's conjecture===
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| [[Vinogradov's theorem]] proves [[Goldbach's weak conjecture]] for sufficiently large ''n''. [[Jean-Marc Deshouillers|Deshouillers]], Effinger, te Riele and Zinoviev conditionally proved the weak conjecture under the [[Generalized Riemann Hypothesis|GRH]].<ref name="DETZ97">Deshouillers, Effinger, Te Riele and Zinoviev, "[http://www.ams.org/era/1997-03-15/S1079-6762-97-00031-0/S1079-6762-97-00031-0.pdf A complete Vinogradov 3-primes theorem under the Riemann hypothesis]", ''Electronic Research Announcements of the American Mathematical Society'' '''3''', pp. 99-104 (1997).</ref> The weak conjecture is known to hold for all ''n'' outside the range <math>(10^{20}, e^{3100}).</math><ref name="DETZ97" /><ref>{{cite journal |first=M. C. |last=Liu |first2=T. Z. |last2=Wang |title=On the Vinogradov bound in the three primes Goldbach conjecture |journal=[[Acta Arithmetica]] |volume=105 |issue= |year=2002 |pages=133–175 |doi=10.4064/aa105-2-3 }}</ref>
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| [[Chen's theorem]] proves that for all sufficiently large ''n'', <math>2n=p+q</math> where ''p'' is prime and ''q'' is either prime or [[semiprime]]. [[Hugh Montgomery (mathematician)|Montgomery]] and [[Robert Charles Vaughan (mathematician)|Vaughan]] showed that the exceptional set (even numbers not expressible as the sum of two primes) was of [[Natural density|density]] zero.<ref>{{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= }}</ref>
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| ===Twin prime conjecture===
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| [[Daniel Goldston|Goldston]], [[János Pintz|Pintz]] and [[Cem Yıldırım|Yıldırım]] showed that the size of the gap between primes could be far smaller than the average size of the [[prime gap]]:
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| :<math>\liminf\frac{p_{n+1}-p_n}{\sqrt{\log p_n}(\log\log p_n)^2}<\infty.</math><ref>Daniel Alan Goldston, Yoichi Motohashi, János Pintz and Cem Yalçın Yıldırım, [http://xxx.lanl.gov/pdf/0710.2728 Primes in tuples. II]. Preprint.</ref>
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| Earlier, they conditionally proved a weaker version of the twin prime conjecture, that infinitely many primes ''p'' exist with <math>\pi(p+20)-\pi(p)\ge1</math>, under the [[Elliott–Halberstam conjecture]].<ref>Daniel Alan Goldston, Yoichi Motohashi, János Pintz and Cem Yalçın Yıldırım, [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pja/1146576181 Small Gaps between Primes Exist]. '' Proceedings of the Japan Academy, Series A Mathematical Sciences'' '''82''' 4 (2006), pp. 61-65.</ref> <math>\pi(x)</math> is the [[prime-counting function]]. The twin prime conjecture replaces 20 with 2.
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| [[Chen Jingrun|Chen]] showed that there are infinitely many primes ''p'' (later called [[Chen prime]]s) such that ''p''+2 is either a prime or a semiprime.
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| [[Yitang Zhang]] showed that there are infinitely many prime pairs with gap bounded by 70 million. This is the first unconditional finite bound on prime gaps.
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| ===Legendre's conjecture===
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| It suffices to check that each prime gap starting at ''p'' is smaller than <math>2\sqrt p.</math> A table of maximal prime gaps shows that the conjecture holds to 10<sup>18</sup>.<ref>Jens Kruse Andersen, [http://users.cybercity.dk/~dsl522332/math/primegaps/maximal.htm Maximal Prime Gaps]</ref> A counterexample near 10<sup>18</sup> would require a prime gap fifty million times the size of the average gap. Matomäki shows that there are at most <math>x^{1/6}</math> exceptional primes followed by gaps larger than <math>\sqrt{2p}</math>; in particular,
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| :<math>\sum_{\stackrel{p_{n+1}-p_n>x^{1/2}}{x\le p_n\le 2x}}p_{n+1}-p_n\ll x^{2/3}.</math><ref>{{cite journal|author=Kaisa Matomäki|title=Large differences between consecutive primes|journal=Quarterly Journal of Mathematics|volume=58|year=2007|pages=pp. 489–518}}.</ref>
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| A result due to [[Albert Ingham|Ingham]] shows that there is a prime between <math>n^3</math> and <math>(n+1)^3</math> for every large enough ''n''.<ref>{{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 }}</ref>
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| ===Near-square primes===
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| The [[Friedlander–Iwaniec theorem]] shows that infinitely many primes are of the form <math>x^2+y^4</math>.
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| [[Henryk Iwaniec|Iwaniec]] showed that there are infinitely many numbers of the form <math>n^2+1</math> with at most two prime factors.<ref>{{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 }}</ref>
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| Todd showed<ref>{{citation|name=J. Todd|title=A problem on arc tangent relations|journal=American Mathematical Monthly|volume=56|year=1949|pages=pp. 517-528}}</ref> that there are infinitely many numbers of the form <math>n^2+1</math> with greatest prime factor at least 2''n''. Replacing 2''n'' with <math>n^2</math> would yield the conjecture.
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| Banks, Friedlander, Pomerance, & Shparlinski show that there are infinitely many squarefree ''n'' for which φ(''n'') is a square. Replacing "squarefree" with "prime" would yield the conjecture.
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| ==Notes==
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| {{Reflist}}
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| ==External links==
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| *{{MathWorld|urlname=LandausProblems|title=Landau's Problems}}
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| [[Category:Conjectures about prime numbers]]
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| [[Category:Unsolved problems in mathematics]]
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