# number theory

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## number theory,

branch of mathematics concerned with the properties of the integers (the numbers 0, 1, −1, 2, −2, 3, −3, …). An important area in number theory is the analysis of prime numbers. A prime number is an integer p>1 divisible only by 1 and p; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that have other divisors are called composite; examples are 4, 6, 8, 9, 10, 12, … . The fundamental theorem of arithmetic, the unique factorization theorem, asserts that any positive integer a is a product (a = p1 · p2 · p3 · · · pn) of primes that are unique except for the order in which they are listed; e.g., the number 20 is the product 20 = 2 · 2 ·5, and it is unique (disregarding order) since 20 has this and only this product of primes. This theorem was known to the Greek mathematician Euclid, who proved that there are infinitely many primes. Analytic number theory has given a further refinement of Euclid's theorem by determining a function that measures how densely the primes are distributed among all integers. Twin primes are primes having a difference of 2, such as (3,5) and (11,13). The modern theory of numbers made its first great advances through the work of Leonhard EulerEuler, Leonhard
, 1707–83, Swiss mathematician. Born and educated at Basel, where he knew the Bernoullis, he went to St. Petersburg (1727) at the invitation of Catherine I, becoming professor of mathematics there on the departure of Daniel Bernoulli (1733).
, C. F. GaussGauss, Carl Friedrich
, born Johann Friederich Carl Gauss, 1777–1855, German mathematician, physicist, and astronomer. Gauss was educated at the Caroline College, Brunswick, and the Univ.
, and Pierre de FermatFermat, Pierre de
, 1601–65, French mathematician. A magistrate whose avocation was mathematics, Fermat is known as a founder of modern number theory and probability theory.
. It remains a major area of mathematical research, to which the most sophisticated mathematical tools have been applied.

### Bibliography

See O. Ore, Number Theory and Its History (1988); R. P. Burn, A Pathway into Number Theory (2d ed. 1996); J. H. Silverman, A Friendly Introduction to Number Theory (1996); M. A. Herkommer, Number Theory: A Programmer's Guide (1998); R. A. Mollin, Algebraic Number Theory (1999).

## number theory

[′nəm·bər ′thē·ə·rē]
(mathematics)
The study of integers and relations between them.
References in periodicals archive ?
GCD is well known elementary number theory algorithm that is defined as the largest positive integer number that can divide two non-negative integer numbers without reminder.
Pollack introduces algebraic number theory to readers who are familiar with linear algebra, commutative ring theory, Galois theory, a little abelian groups theory, and elementary number theory up to and including the law of quadratic reciprocity.
It's a book that will work well with most math or computing science courses, on a subject that pertains to graph theory, point set topology, elementary number theory, linear algebra, analysis, probability theory, geometry, group theory, and game theory, among many other topics.
Topics include Ramsey number theory (that there cannot be complete disorder and in any large system there must always be some structure), additive number theory, multiplicative number theory, combinatorial games, sequences, elementary number theory and graph theory.
New sections on applications of elementary number theory, multidimensional induction, counting tulips, and the binomial distribution
With an emphasis of proofs, this book explains concepts in advanced mathematics: the elements of logic, techniques of proof, elementary theory of sets, functions, relations, axiomatic theory of positive integers, elementary number theory, cardinality, counting techniques and combinatorics, and the construction of integers, rationals, and real and complex numbers.
He assumes readers know the basics of elementary number theory such as divisibility, primes, quadratic reciprocity, and the representation of integers by binary quadratic forms.
The 23 contributions address such topics as how students develop elementary number theory concepts when using calculators; the use of spreadsheet software to connect proportional reasoning to the real world in a middle school classroom; and ways of engaging students in mathematics activities through calculators and small robots.
In elementary number theory, we call an arithmetical function f (n) as an additive function, if for any positive integers m, n with (m, n) = 1, we have f (mn) = f (m) + f (n).
In elementary number theory, there are many arithmetical functions satisfying the additive properties.
Then from the elementary number theory textbook (see [6], [7]) and reference [8] we know that
6] Chengdong Pan and Chengbiao Pan, The Elementary Number Theory, Beijing University Press, Beijing, 2003.

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