A **Diophantine quintuple** is a set of five positive integers \( (a, b, c, d, e) \) such that the sum of any two distinct elements in the set is a perfect square.

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Diophantus II.VIII refers to a specific problem in the ancient Greek mathematician Diophantus's work, "Arithmetica." This text is one of the earliest known to study algebraic equations and includes numerous problems that focus on finding integer solutions to polynomial equations. In this specific section, Diophantus presents a problem involving the search for rational (or integer) solutions to a particular equation.

Diophantus of Alexandria was a Greek mathematician who lived around the 3rd century AD. He is best known for his work in number theory, particularly for his contributions to what are now known as Diophantine equations. His most famous work is the "Arithmetica," where he introduced methods for solving equations that require integer solutions. **Diophantine Equations** are polynomial equations that seek integer solutions.

In number theory, "effective results" refer to theorems or results that not only provide qualitative information (e.g., existence, properties, etc.) about mathematical objects but also yield explicit methods, algorithms, or bounds that allow for the computation of specific examples or the verification of claims. Essentially, an effective result provides a concrete way to achieve or demonstrate what a more abstract result asserts.

The Erdős–Moser equation is a specific type of functional equation that arises in the context of additive combinatorics and related fields in mathematics. The equation is named after Paul Erdős and Leo Moser, who studied its properties.

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The Erdős–Straus conjecture is a problem in number theory that was proposed by the mathematicians Paul Erdős and George Strauss in 1948. The conjecture asserts that for every integer \( n \geq 2 \), the equation \[ \frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \] has solutions in positive integers \( x, y, z \).

An Euler brick is a special type of rectangular cuboid (or box) with integer side lengths \(a\), \(b\), and \(c\) such that the lengths of the three face diagonals are also integers. Specifically, the conditions for an Euler brick are that: 1. The dimensions are positive integers: \(a\), \(b\), and \(c\). 2. The lengths of the face diagonals are also integers.

Fermat's right triangle theorem states that if \( a \), \( b \), and \( c \) are the lengths of the sides of a right triangle, with \( c \) being the length of the hypotenuse, then the only integer solutions to the equation \( a^2 + b^2 = c^2 \) occur for certain sets of values for \( a \), \( b \), and \( c \).

The Goormaghtigh conjecture is a hypothesis in the field of number theory, specifically concerning the distribution of prime numbers and their relationship with integers. Proposed by the Belgian mathematician Louis Goormaghtigh in the early 20th century, the conjecture states that there are infinitely many prime numbers \( p \) such that \( p + 1 \) is a perfect square.

Hilbert's tenth problem, proposed by mathematician David Hilbert in 1900, asks for an algorithm that, given a polynomial Diophantine equation—a polynomial equation where the variables are to be integers—will determine whether there are any integer solutions to that equation.

Hippolyte Sébron (1821–1879) was a French painter known for his contributions to the art world during the 19th century. He was primarily associated with the realist movement and is often recognized for his detailed and vibrant landscape paintings, as well as portraits and historical scenes. Sébron was influenced by the naturalistic approach of his contemporaries, and his works often capture the essence of French rural life and the beauty of the countryside.

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The Jacobi–Madden equation refers to a mathematical relationship that arises in the context of dynamics, particularly in the study of second-order equations and Hamiltonian mechanics. It is associated with the properties and transformations of certain integrable systems.

The Lander, Parkin, and Selfridge conjecture is a statement in number theory that pertains to the existence of certain types of prime numbers. Specifically, it deals with prime numbers that can be represented in a specific way using two distinct primes.

Legendre's equation, often encountered in mathematical physics and potential theory, refers to a specific type of ordinary differential equation. It is given in the context of Legendre polynomials, which are solutions to this equation.

Leonid Levin is a prominent computer scientist known for his significant contributions to computational complexity theory, algorithms, and computer science in general. He was born in 1948 in the former Soviet Union and later emigrated to the United States. Levin is particularly known for his work on NP-completeness and for his contributions to the theory of randomized algorithms.

A Markov number is a specific type of positive integer that is associated with a particular solution to Markov's equation, which is given by: \[ x^2 + y^2 + z^2 = 3xyz \] where \( x \), \( y \), and \( z \) are positive integers. A set of numbers \( (x, y, z) \) that satisfies this equation is called a Markov triple.