Diophantine equation
In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such tha...
Diophantine equation - Wikipedia
Bézout's identity
Bézout's identity (also called Bézout's lemma) is a theorem in the elementary theory of numbers: let a and b be nonzero integers and let d be their greatest common divisor. Then there exist integers x...
Bézout's identity - Wikipedia
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not...
Tree of Pythagorean triples
In mathematics, a Pythagorean triple is a set of three positive integers a, b, and c having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfy...
Tree of Pythagorean triples - Wikipedia
Lander, Parkin, and Selfridge conjecture
The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theo...
Lander, Parkin, and Selfridge conjecture - Wikipedia
Effective results in number theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if th...
Tsen rank
In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their...
Semistable abelian variety
In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field.For an abelian varie...
Brahmagupta's problem
This problem was given in India by the mathematician Brahmagupta in 628 in his treatise Brahma Sputa Siddhanta:solve the Pell equationBrahmagupta gave the smallest solution as
Brauer's theorem on forms
In mathematics, Brauer's theorem, named for Richard Brauer, is a result on the representability of 0 by forms over certain fields in sufficiently many variables.
Let K be a field such that for eve...
Siegel's lemma
In transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence o...
Thue equation
In mathematics, a Thue equation is a Diophantine equation of the formwhere ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number. It is ...
Congruent number
In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with t...
Lonely runner conjecture
In number theory, and especially the study of diophantine approximation, the lonely runner conjecture is a conjecture originally due to J. M. Wills in 1967. Applications of the conjecture are widespr...
Lonely runner conjecture - Wikipedia
Hasse principle
In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to pie...
Hilbert's tenth problem
Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900. Its statement is as follows:It took many years for the problem to be solved with a negative answer. Today, it is known t...
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 ...
Integer triangle
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rat...
Euler's sum of powers conjecture
Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the...
Erdős–Straus conjecture
In number theory, the Erdős–Straus conjecture states that for all integers n ≥ 2, the rational number 4/n can be expressed as the sum of three unit fractions. Paul Erdős and Ernst G. Straus formulate...
Chakravala method
The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 118...
Infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the facts that the natural numbers are well ordered and that there are only a finite number o...
Archimedes' cattle problem
Archimedes' cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, th...
Jacobi–Madden equation
The Jacobi–Madden equation is a Diophantine equationproposed by the physicist Lee W. Jacobi and the mathematician Daniel J. Madden in 2008. The variables a, b, c, and d can be any integers, positive, ...
Jacobi–Madden equation - Wikipedia
Euler brick
In mathematics, an Euler brick, named after Leonhard Euler, is a cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relati...
Robbins pentagon
In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers.
Robbins pentagons were named by Buchholz & MacDougall (2008) after David P. Robbins, ...
Robbins pentagon - Wikipedia
Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that a + b = c. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythago...
Pythagorean triple - Wikipedia