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At the 1912
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...
,
Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopo ...
listed four basic problems about
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s. 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: #
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the ...
: Can every even
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
greater than 2 be written as the sum of two primes? # Twin prime conjecture: Are there infinitely many primes ''p'' such that ''p'' + 2 is prime? # Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares? # 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''2 + 1? , all four problems are unresolved.


Progress toward solutions


Goldbach's conjecture

Goldbach's weak conjecture In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that : Every odd number greater than 5 can be expressed as the sum of three prime number, prime ...
, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the ...
. Ivan Vinogradov proved it for large enough ''n'' ( Vinogradov's theorem) in 1937, and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013.
Chen's theorem In number theory, Chen's theorem states that every sufficiently large parity (mathematics), even number can be written as the sum of either two prime number, primes, or a prime and a semiprime (the product of two primes). It is a weakened form o ...
, another weakening of Goldbach's conjecture, proves that for all sufficiently large ''n'', 2n=p+q where ''p'' is prime and ''q'' is either prime or semiprime.A semiprime is a natural number that is the product of two prime factors. Bordignon, Johnston, and Starichkova, correcting and improving on Yamada, proved an explicit version of Chen's theorem: every even number greater than e^ \approx 1.4\cdot10^ is the sum of a prime and a product of at most two primes. Bordignon and Starichkova reduce this to e^ \approx 3.6\cdot10^ assuming the
Generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whi ...
(GRH) for
Dirichlet L-function In mathematics, a Dirichlet L-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and s a complex variable with real part greater than 1 . It is a special case of a Dirichlet series. By anal ...
s. Johnston and Starichkova give a version working for all ''n'' ≥ 4 at the cost of using a number which is the product of at most 369 primes rather than a prime or semiprime; under GRH they improve 369 to 33. Montgomery and
Vaughan Vaughan ( ) (2022 population 344,412) is a city in Ontario, Canada. It is located in the Regional Municipality of York, just north of Toronto. Vaughan was the fastest-growing municipality in Canada between 1996 and 2006 with its population increa ...
showed that the exceptional set of even numbers not expressible as the sum of two primes has a
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
zero, although the set is not proven to be finite. The best current bounds on the exceptional set is E(x) < x^ (for large enough ''x'') due to Pintz, and E(x) \ll x^\log^3 x under RH, due to Goldston. 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. Following many advances (see Pintz for an overview), Pintz and Ruzsa improved this to ''K'' = 8. Assuming the GRH, this can be improved to ''K'' = 7.


Twin prime conjecture

In 2013 Yitang Zhang 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 The Polymath Project is a collaboration among mathematicians to solve important and difficult mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution. The project began in J ...
. Under the generalized Elliott–Halberstam conjecture this was improved to 6, extending earlier work by Maynard and Goldston, Pintz and Yıldırım. In 1966 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 2 \sqrt p. A table of maximal prime gaps shows that the
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
holds to 264 ≈ 1.8. A
counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...
near that size would require a prime gap a hundred million times the size of the average gap. Järviniemi, improving on work by Heath-Brown and by Matomäki, shows that there are at most x^ exceptional primes followed by gaps larger than \sqrt; in particular, :\sum_p_-p_n\ll x^. A result due to Ingham shows that there is a prime between n^3 and (n+1)^3 for every large enough ''n''.


Near-square primes

Landau's fourth problem asked whether there are infinitely many primes which are of the form p=n^2+1 for integer ''n''. (The list of known primes of this form is .) 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. One example of near-square primes are Fermat primes. Henryk Iwaniec showed that there are infinitely many numbers of the form n^2+1 with at most two prime factors. Ankeny and Kubilius proved that, assuming the extended Riemann hypothesis for ''L''-functions on Hecke characters, there are infinitely many primes of the form p=x^2+y^2 with y=O(\log p). Landau's conjecture is for the stronger y=1. The best unconditional result is due to Harman and Lewis and it gives y=O(p^). Grimmelt & Merikoski, improving on previous works, showed that there are infinitely many numbers of the form n^2+1 with greatest prime factor at least n^. Replacing the exponent with 2 would yield Landau's conjecture. The Friedlander–Iwaniec theorem shows that infinitely many primes are of the form x^2+y^4. Baier and Zhao prove that there are infinitely many primes of the form p=an^2+1 with a < p^; the exponent can be improved to 1/2+\varepsilon under the Generalized Riemann Hypothesis for L-functions and to \varepsilon under a certain Elliott-Halberstam type hypothesis. The Brun sieve establishes an upper bound on the density of primes having the form p=n^2+1: there are O(\sqrt x/\log x) such primes up to x. Hence
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
numbers of the form n^2+1 are composite.


See also

*
List of unsolved problems in mathematics Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, Mathematical analysis, analysis, combinatorics, Algebraic geometry, alge ...
*
Hilbert's problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pr ...


Notes


References


External links

* {{Prime number conjectures Conjectures about prime numbers Unsolved problems in number theory