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Probabilistic number theory
 
 
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Probabilistic number theory is a subfield of number theory
Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, which explicitly uses probability
Probability

Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
 to answer questions of number theory. One basic idea underlying it is that different prime number
Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC....
s are, in some serious sense, like independent random variables. This however is not an idea that has a unique useful formal expression.

The founders of the theory were Paul Erdos
Paul Erdos

Paul Erdos was an immensely prolific and famously eccentric Hungary mathematician. With hundreds of collaborators, he worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory....
, Aurel Wintner and Mark Kac
Mark Kac

Mark Kac was a Poles and United States mathematician of Jewish ancestry. His main interest was probability theory. His question, "Hearing the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry....
 during the 1930s, one of the most intense periods of investigation in analytic number theory
Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve number-theoretical problems....
.






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Encyclopedia


Probabilistic number theory is a subfield of number theory
Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, which explicitly uses probability
Probability

Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
 to answer questions of number theory. One basic idea underlying it is that different prime number
Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC....
s are, in some serious sense, like independent random variables. This however is not an idea that has a unique useful formal expression.

The founders of the theory were Paul Erdos
Paul Erdos

Paul Erdos was an immensely prolific and famously eccentric Hungary mathematician. With hundreds of collaborators, he worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory....
, Aurel Wintner and Mark Kac
Mark Kac

Mark Kac was a Poles and United States mathematician of Jewish ancestry. His main interest was probability theory. His question, "Hearing the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry....
 during the 1930s, one of the most intense periods of investigation in analytic number theory
Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve number-theoretical problems....
. The Erdos-Wintner theorem on additive function
Additive function

Different definitions exist depending on the specific field of application. Traditionally, an additive function is a function that preserves the addition operation:for any two elements x and y in the domain....
s was a foundational result.

See also


  • analytic number theory
    Analytic number theory

    In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve number-theoretical problems....
  • areas of mathematics
    Areas of mathematics

    Here is a list of areas of modern mathematics, with a brief explanation of their scope and links to other parts of this encyclopedia, set out in a systematic way....
  • list of number theory topics
    List of number theory topics

    This is a list of number theory topics, by Wikipedia page. See also*List of recreational number theory topics*Topics in cryptography...
  • list of probability topics
    List of probability topics

    This is a list of probability topics, by Wikipedia page.It overlaps with the list of statistical topics. There are also the Catalog of articles in probability theory, list of probabilists and list of statisticians....
  • mathematics
    Mathematics

    Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
  • probabilistic method
    Probabilistic method

    The probabilistic method is a nonconstructive proof method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object....
  • probable prime
    Probable prime

    In number theory, a probable prime is an integer that satisfies a specific condition also satisfied by all prime numbers. Different types of probable primes have different specific conditions....