
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'',
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
is the sum of a prime and a product of at most two primes. Bordignon and Starichkova reduce this to
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
(for large enough ''x'') due to
Pintz, and
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
. 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 2
64 ≈ 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
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''.
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 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
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
with
. Landau's conjecture is for the stronger
. The best unconditional result is due to Harman and Lewis and it gives
.
Grimmelt & Merikoski, improving on previous works, 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
Friedlander–Iwaniec theorem shows that infinitely many primes are of the form
.
Baier and Zhao
prove that there are infinitely many primes of the form
with
; the exponent can be improved to
under the Generalized Riemann Hypothesis for L-functions and to
under a certain Elliott-Halberstam type hypothesis.
The
Brun sieve establishes an upper bound on the density of primes having the form
: there are
such primes up to
. 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
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