Category: Diophantine equations

Tijdeman's theorem
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 xy = yx
In general, exponentiation fails to be commutative. However, the equation holds in special cases, such as
Hundred Fowls Problem
The Hundred Fowls Problem is a problem first discussed in the fifth century CE Chinese mathematics text Zhang Qiujian suanjing (The Mathematical Classic of Zhang Qiujian), a book of mathematical probl
Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem: Bézout's identity — Let a and b be integers with greatest common divisor d. Then t
Sum of four cubes problem
The sum of four cubes problem asks whether every integer is the sum of four cubes of integers. It is conjectured the answer is affirmative, but this conjecture has been neither proved nor disproved. S
Beal conjecture
The Beal conjecture is the following conjecture in number theory: Unsolved problem in mathematics: If where A, B, C, x, y, z are positive integers and x, y, z are ≥ 3, do A, B, and C have a common pri
Diophantine quintuple
In mathematics, a diophantine m-tuple is a set of m positive integers such that is a perfect square for any . A set of m positive rational numbers with the similar property that the product of any two
Fermat–Catalan conjecture
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many
Goormaghtigh conjecture
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
Heegner's lemma
In mathematics, Heegner's lemma is a lemma used by Kurt Heegner in his paper on the class number problem. His lemma states that if is a curve over a field with a4 not a square, then it has a solution
Diophantus and Diophantine Equations
Diophantus and Diophantine Equations is a book in the history of mathematics, on the history of Diophantine equations and their solution by Diophantus of Alexandria. It was originally written in Russi
Lonely runner conjecture
In number theory, specifically the study of Diophantine approximation, the lonely runner conjecture is a conjecture about the long-term behavior of runners on a circular track. It states that runners
Proof by infinite descent
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number,
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
The monkey and the coconuts
The monkey and the coconuts is a mathematical puzzle in the field of Diophantine analysis that originated in a magazine fictional short story involving five sailors and a monkey on a desert island who
Legendre's equation
In mathematics, Legendre's equation is the Diophantine equation The equation is named for Adrien-Marie Legendre who proved in 1785 that it is solvable in integers x, y, z, not all zero, if and only if
Prouhet–Tarry–Escott problem
In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials are all equal.That is, the two multisets shou
Tunnell's theorem
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.
Siegel identity
In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations.
Eisenstein triple
Similar to a Pythagorean triple, an Eisenstein triple (named after Gotthold Eisenstein) is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 or 120 degre
Coin problem
The coin problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that asks for the largest monetary amount
Diophantus II.VIII
The eighth problem of the second book of Arithmetica by Diophantus (c. 200/214 AD – c. 284/298 AD) is to divide a square into a sum of two squares.
Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pyth
Bhaskara's lemma
Bhaskara's Lemma is an identity used as a lemma during the chakravala method. It states that: for integers and non-zero integer .
ABS methods
ABS methods, where the acronym contains the initials of Jozsef Abaffy, Charles G. Broyden and Emilio Spedicato, have been developed since 1981 to generate a large class of algorithms for the following
Markov number
A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation studied by Andrey Markoff . The first few Markov numbers are 1, 2, 5, 13
Mordell curve
In algebra, a Mordell curve is an elliptic curve of the form y2 = x3 + n, where n is a fixed non-zero integer. These curves were closely studied by Louis Mordell, from the point of view of determining
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 – 1185 CE
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 Theor
Perfect digital invariant
In number theory, a perfect digital invariant (PDI) is a number in a given number base that is the sum of its own digits each raised to a given power.
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
Jacobi–Madden equation
The Jacobi–Madden equation is the Diophantine equation proposed 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, positiv
Erdős–Moser equation
In number theory, the Erdős–Moser equation is where and are positive integers. The only known solution is 11 + 21 = 31, and Paul Erdős conjectured that no further solutions exist.
Cannonball problem
In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for wha
Cuboid conjectures
No description available.
Pythagorean quadruple
A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a2 + b2 + c2 = d2. They are solutions of a Diophantine equation and often only positive integer values are considered. However,
Norm form
In mathematics, a norm form is a homogeneous form in n variables constructed from the field norm of a field extension L/K of degree n. That is, writing N for the norm mapping to K, and selecting a bas
Tree of primitive Pythagorean triples
In mathematics, a tree of primitive Pythagorean triples is a data tree in which each node branches to three subsequent nodes with the infinite set of all nodes giving all (and only) primitive Pythagor
Euler brick
In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge length
Hilbert's tenth problem
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 which, for any g
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, where x1, .
Erdős–Diophantine graph
An Erdős–Diophantine graph is an object in the mathematical subject of Diophantine equations consisting of a set of integer points at integer distances in the plane that cannot be extended by any addi
Archimedes's cattle problem
Archimedes's 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, t
Brown numbers
No description available.
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones.
Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian
Vieta jumping
In number theory, Vieta jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two positive integers is given, along with a statemen
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.
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
Birch's theorem
In mathematics, Birch's theorem, named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.
In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares.That is, if , , and (for integers , , and ) are three sq
Fermat's right triangle theorem
Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. I
List of sums of reciprocals
In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the su
Thue equation
In mathematics, a Thue equation is a Diophantine equation of the form ƒ(x,y) = r, where ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational n
Siegel's theorem on integral points
In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there a
Sums of three cubes
In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum.
Brahmagupta's problem
This problem was given in India by the mathematician Brahmagupta in 628 AD in his treatise Brahma Sputa Siddhanta: Solve the Pell's equation for integers . Brahmagupta gave the smallest solution as .
Kaprekar number
In mathematics, a natural number in a given number base is a -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add u
Catalan's conjecture
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderbor
Ramanujan–Nagell equation
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 exponent
Brocard's problem
Brocard's problem is a problem in mathematics that asks to find integer values of and for which where is the factorial. It was posed by Henri Brocard in a pair of articles in 1876 and 1885, and indepe
Erdős–Straus conjecture
Unsolved problem in mathematics: Does have a positive integer solution for every integer ? (more unsolved problems in mathematics) The Erdős–Straus conjecture is an unproven statement in number theory
Optic equation
In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers a and b to equal the reciprocal of a third positive integer c: Multiplying both si