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In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements . It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers, or defined as generalizations of the integers.
In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.
A prime number is a natural number greater than 1 that is not a 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 because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4.
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation, can decide whether the equation has a solution with all unknowns taking integer values.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals.
In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.
In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalization of widely open conjectures such as the twin prime conjecture. The hypothesis is named after Andrzej Schinzel.
In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation
Robert Tijdeman is a Dutch mathematician. Specializing in number theory, he is best known for his Tijdeman's theorem. He is a professor of mathematics at the Leiden University since 1975, and was chairman of the department of mathematics and computer science at Leiden from 1991 to 1993. He was also president of the Dutch Mathematical Society from 1984 to 1986.
In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form
This is a glossary of concepts and results in number theory, a field of mathematics. Concepts and results in arithmetic geometry and diophantine geometry can be found in Glossary of arithmetic and diophantine geometry.
References
- ↑ Sukumar Mallick; Saguna Dewan; S C Dhawan (1999). Handbook of Shanti Swarup Bhatnagar Prize Winners(1958 - 1998) (PDF). New Delhi: Human Rsource Development Group, Council of Scientific & Industrial Research. p. 118.
- ↑ Filaseta, Michael; Carrie Finch; J Russell Leidy (2008). "T. N. Shorey's Influence in the Theory of Irreducible Polynomials". Diophantine Equations (Ed. N. Saradha). New Delhi: Narosa Publ. House.
