The term "Optic equation" does not refer to a specific, universally recognized equation in optics. Instead, it may refer to several key equations and principles used in the field of optics, which is the study of light and its behavior. 1. **Lens Maker's Equation**: This equation relates the focal length of a lens to the radii of curvature of its two surfaces and the refractive index of the lens material.

A Mordell curve is a type of algebraic curve defined by a specific type of equation. More formally, it can be described as an elliptic curve given by a Weierstrass equation of the form: \[ y^2 = x^3 + k \] where \( k \) is a constant. These curves are named after the mathematician Louise Mordell, who studied the properties of such equations and their rational points.

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.

The Lonely Runner Conjecture is a hypothesis in the field of number theory and combinatorial geometry. It proposes that if \( k \) runners, each moving at different constant speeds, start running around a circular track of unit length, then for sufficiently large time, each runner will be at a distance of at least \( \frac{1}{k} \) from every other runner at some point in time.

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.

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.

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.

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.

Heegner's lemma is a result in number theory that is primarily concerned with the representation of integers as sums of squares. It plays an important role in the theory of quadratic forms and has implications in the study of class numbers and other aspects of algebraic number theory. Specifically, Heegner's lemma provides a condition under which certain integers can be represented as sums of two squares.

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.

The Fermat–Catalan conjecture is a conjecture in number theory that deals with a specific type of equation related to powers of 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 \( c \).

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.

Euler's sum of powers conjecture is a proposition made by the mathematician Leonhard Euler in the 18th century. It suggests a relationship between sums of powers of natural numbers and the need for certain numbers to be larger than expected to represent these sums as higher-order powers. The conjecture is specifically about the representation of numbers as sums of n-th powers of integers.

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 \( x, y, z \).

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.

The equation \( xy = yx \) describes a relationship between the variables \( x \) and \( y \). It essentially states that the product of \( x \) and \( y \) is equal to the product of \( y \) and \( x \). This equation holds true for any real numbers \( x \) and \( y \) due to the commutative property of multiplication, which states that the order of multiplication does not affect the result.

An **Eisenstein triple** is a concept from number theory that refers to a specific type of three-tuple of integers (a, b, c) that satisfies certain conditions related to Eisenstein integers.

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.

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.