diophantine equation


diophantine equation
/duy'euh fan"tuyn, -teen, -fan"tn/, Math.
an equation involving more than one variable in which the coefficients of the variables are integers and for which integral solutions are sought.
[1925-30; named after Diophantus, 3rd-century A.D. Greek mathematician; see -INE1]

* * *

      equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers. For example, 3x + 7y = 1 or x2 − y2 = z3, where x, y, and z are integers. Named in honour of the 3rd-century Greek mathematician Diophantus of Alexandria, these equations were first systematically solved by Hindu mathematicians beginning with Āryabhaṭa I (Aryabhata I) (c. 476–550).

      Diophantine equations fall into three classes: those with no solutions, those with only finitely many solutions, and those with infinitely many solutions. For example, the equation 6x − 9y = 29 has no solutions, but the equation 6x − 9y = 30, which upon division by 3 reduces to 2x − 3y = 10, has infinitely many. For example, x = 20, y = 10 is a solution, and so is x = 20 + 3t, y = 10 + 2t for every integer t, positive, negative, or zero. This is called a one-parameter family of solutions, with t being the arbitrary parameter.

      Congruence methods provide a useful tool in determining the number of solutions to a Diophantine equation. Applied to the simplest Diophantine equation, ax + by = c, where a, b, and c are nonzero integers, these methods show that the equation has either no solutions or infinitely many, according to whether the greatest common divisor (GCD) of a and b divides c: if not, there are no solutions; if it does, there are infinitely many solutions, and they form a one-parameter family of solutions.

* * *


Universalium. 2010.

Look at other dictionaries:

  • Diophantine equation — In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for… …   Wikipedia

  • Diophantine equation — noun Etymology: Diophantus, 3d century A.D. Greek mathematician Date: circa 1928 an indeterminate polynomial equation which has integral coefficients and for which it is required to find all integral solutions …   New Collegiate Dictionary

  • Diophantine equation — noun A polynomial equation whose variables are only permitted to assume integer values …   Wiktionary

  • Diophantine equation — [ˌdʌɪə fantɪn, tʌɪn] noun Mathematics a polynomial equation with integral coefficients for which integral solutions are required. Origin C18: named after the third cent. Greek mathematician Diophantus …   English new terms dictionary

  • Diophantine equation — /daɪəfæntaɪn əˈkweɪʒən/ (say duyuhfantuyn uh kwayzhuhn) noun an indeterminate equation with integral coefficients, in which the variables can take only integral values. {from Diophantus, fl. 3rd century AD, Greek mathematician} …   Australian English dictionary

  • diophantine equation — n. Math. an equation with integral coefficients for which integral solutions are required. Etymology: Diophantus of Alexandria, mathematician of uncert. date …   Useful english dictionary

  • Polynomial Diophantine equation — In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation whose solutions are restricted to be polynomials in the indeterminate. A Diophantine equation, in general, is one where the solutions are restricted to some …   Wikipedia

  • Equation diophantienne — Équation diophantienne Édition de 1670 des Arithmétiques de Diophante d Alexandrie. Une équation diophantienne, en mathématiques, est une équation dont les coefficients sont des nombres entiers et dont les solutions recherchées sont également… …   Wikipédia en Français

  • Diophantine — means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: Diophantine approximation Diophantine equation Diophantine set This disambiguation page lists articles associated with the same title. If an …   Wikipedia

  • Diophantine set — In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk)=0 (usually abbreviated P(x,y)=0 ) where P(x,y) is a polynomial with integer coefficients. A Diophantine set is a subset S of Nj [1] so that for some… …   Wikipedia