Graphical proof for Andrica's conjecture for the first (a)100, (b)200 and (c)500 prime numbers. It is conjectured that the function is always less than 1.
Andrica's conjecture (named after Romanian mathematician Dorin Andrica (es)) is a conjecture regarding the gaps between prime numbers.[1]
holds for all , where is the nth prime number. If denotes the nth prime gap, then Andrica's conjecture can also be rewritten as
Empirical evidence
Imran Ghory has used data on the largest prime gaps to confirm the conjecture for up to 1.3002×1016.[2] Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4×1018.
The discrete function is plotted in the figures opposite. The high-water marks for occur for n=1,2, and4, with A4 ≈ 0.670873..., with no larger value among the first 105 primes. Since the Andrica function decreases asymptotically as n increases, a prime gap of ever increasing size is needed to make the difference large as n becomes large. It therefore seems highly likely the conjecture is true, although this has not yet been proven.
Generalizations
Value of x in the generalized Andrica's conjecture for the first 100 primes, with the conjectured value of xmin labeled.
As a generalization of Andrica's conjecture, the following equation has been considered:
where is the nth prime and x can be any positive number.
The largest possible solution for x is easily seen to occur for n=1, when xmax=1. The smallest solution for x is conjectured to be xmin≈0.567148... (sequence A038458 in the OEIS) which occurs for n=30.
This conjecture has also been stated as an inequality, the generalized Andrica conjecture:
The abc conjecture is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers and that are relatively prime and satisfy . The conjecture essentially states that the product of the distinct prime factors of is usually not much smaller than . A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".
In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers and additive number theory.
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that
In algebra, a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.
In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of Diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938. They also occur in the uniqueness problem for Fourier series. Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s. Salem numbers are a closely related set of numbers.
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between and for every positive integer . The conjecture is one of Landau's problems (1912) on prime numbers, and is one of many open problems on the spacing of prime numbers.
A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation
In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four variables, strengthening his 1924 dissertation research on five or more variables.
In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states:
A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n + 1)-st and the n-th prime numbers, i.e.
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:
Goldbach's conjecture: Can every even integer 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 n2 + 1?
In mathematics, the Hilbert symbol or norm-residue symbol is a function from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by David Hilbert in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.
In mathematics, the Lagrange numbers are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem.
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann, after whom it is named.
Oppermann's conjecture is an unsolved problem in mathematics on the distribution of prime numbers. It is closely related to but stronger than Legendre's conjecture, Andrica's conjecture, and Brocard's conjecture. It is named after Danish mathematician Ludvig Oppermann, who announced it in an unpublished lecture in March 1877.
In number theory, Firoozbakht's conjecture is a conjecture about the distribution of prime numbers. It is named after the Iranian mathematician Farideh Firoozbakht who stated it in 1982.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.
