Graduate Texts in Mathematics 232

... Having spoken of diﬀerent approaches, it will surprise few readers that number theory has many streams. A major surprise is the fact that some of these meet again: Chapter 11 shows that many of the themes in Chapters 1–10 become reconciled further on. The classical class number formula reconciles th ...

... Having spoken of diﬀerent approaches, it will surprise few readers that number theory has many streams. A major surprise is the fact that some of these meet again: Chapter 11 shows that many of the themes in Chapters 1–10 become reconciled further on. The classical class number formula reconciles th ...

Solving Diophantine Equations

... A Diophantine equation is a linear equation ax + by = c where a, b, c ∈ Z and the solutions x and y are also integer numbers. This equation can be completely solved by the well known algorithm proposed by Brahmagupta [Weisstein, 2014b]. In 1900, Hilbert wondered if there is an universal algorithm th ...

... A Diophantine equation is a linear equation ax + by = c where a, b, c ∈ Z and the solutions x and y are also integer numbers. This equation can be completely solved by the well known algorithm proposed by Brahmagupta [Weisstein, 2014b]. In 1900, Hilbert wondered if there is an universal algorithm th ...

Topology of numbers

... As an introduction to the sorts of questions that we will be studying, let us consider right triangles whose sides all have integer lengths. The most familiar example is the (3, 4, 5) right triangle, but there are many others as well, such as the (5, 12, 13) right triangle. Thus we are looking for t ...

... As an introduction to the sorts of questions that we will be studying, let us consider right triangles whose sides all have integer lengths. The most familiar example is the (3, 4, 5) right triangle, but there are many others as well, such as the (5, 12, 13) right triangle. Thus we are looking for t ...

Rational Numbers

... You began your study of numbers by counting objects around you. The numbers used for this purpose were called counting numbers or natural numbers. They are 1, 2, 3, 4, ... By including 0 to natural numbers, we got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural numbers were then pu ...

... You began your study of numbers by counting objects around you. The numbers used for this purpose were called counting numbers or natural numbers. They are 1, 2, 3, 4, ... By including 0 to natural numbers, we got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural numbers were then pu ...

A Combinatorial Miscellany

... respectable than other areas, in spite of many services rendered to both pure and applied mathematics. Then along came the prince of computer science with its many mathematical problems and needs — and it was combinatorics that best fitted the glass slipper held out. The developments within mathemat ...

... respectable than other areas, in spite of many services rendered to both pure and applied mathematics. Then along came the prince of computer science with its many mathematical problems and needs — and it was combinatorics that best fitted the glass slipper held out. The developments within mathemat ...

# Number theory

Number theory (or arithmetic) is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called ""The Queen of Mathematics"" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (Diophantine approximation).The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by ""number theory"". (The word ""arithmetic"" is used by the general public to mean ""elementary calculations""; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.