List of topics named after Leonhard Euler

• Euler's sum of powers conjecture disproved for exponents 4 and 5 during the 20th century; unsolved for higher exponents • Euler's Graeco-Latin square conjecture proved to be true for and disproved otherwise, during the 20th century == Equations ==

Usually, ''Euler's equation'' refers to one of (or a set of) differential equations (DEs). It is customary to classify them into ODEs and PDEs. Otherwise, ''Euler's equation'' may refer to a non-differential equation, as in these three cases: • Euler–Lotka equation, a characteristic equation employed in mathematical demography • Euler's pump and turbine equation • Euler transform used to accelerate the convergence of an alternating series and is also frequently applied to the hypergeometric series Ordinary differential equations • Euler rotation equations, a set of first-order ODEs concerning the rotations of a rigid body. • Euler–Cauchy equation, a linear equidimensional second-order ODE with variable coefficients. Its second-order version can emerge from Laplace's equation in polar coordinates. • Euler–Bernoulli beam equation, a fourth-order ODE concerning the elasticity of structural beams. • Euler's differential equation, a first order nonlinear ordinary differential equation Partial differential equations • Euler conservation equations, a set of quasilinear first-order hyperbolic equations used in fluid dynamics for inviscid flows. In the (Froude) limit of no external field, they are conservation equations. • Euler–Tricomi equation – a second-order PDE emerging from Euler conservation equations. • Euler–Poisson–Darboux equation, a second-order PDE playing important role in solving the wave equation. • Euler–Lagrange equation, a second-order PDE emerging from minimization problems in calculus of variations. • Euler–Arnold equation, describes the evolution of a velocity field when the Lagrangian flow is a geodesic in a group of smooth transformations. == Formulas ==

{{columns-list|colwidth=30em| • Euler's formula, • Euler's polyhedral formula for planar graphs or polyhedra: , a special case of the Euler characteristic in topology • Euler's formula for the critical load of a column: P_\text{cr}=\frac{\pi^2 EI}{(KL)^2} • Euler's continued fraction formula connecting a finite sum of products with a finite continued fraction • Euler product formula for the Riemann zeta function. • Euler–Maclaurin formula (Euler's summation formula) relating integrals to sums • Euler–Rodrigues formula describing the rotation of a vector in three dimensions • Euler's reflection formula, reflection formula for the gamma function • Local Euler characteristic formula }} == Functions ==

• The Euler function, a modular form that is a prototypical q-series. • Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer. • Euler hypergeometric integral • Euler–Riemann zeta function == Identities ==

• Euler's identity . • Euler's four-square identity, which shows that the product of two sums of four squares can itself be expressed as the sum of four squares. • ''Euler's identity'' may also refer to the pentagonal number theorem. == Numbers ==

• Euler's number, e=2.71828\dots, the base of the natural logarithm • Euler's idoneal numbers, a set of 65 or possibly 66 or 67 integers with special properties • Euler numbers, integers occurring in the coefficients of the Taylor series of 1/cosh t • Eulerian numbers count certain types of permutations. • Euler number (physics), the cavitation number in fluid dynamics. • Euler number (algebraic topology) – now, Euler characteristic, classically the number of vertices minus edges plus faces of a polyhedron. • Euler number (3-manifold topology) – see Seifert fiber space • Lucky numbers of Euler • Euler's constant gamma \gamma=0.57721\dots, also known as the Euler–Mascheroni constant • Eulerian integers, more commonly called Eisenstein integers, the algebraic integers of form where is a complex cube root of 1. • Euler–Gompertz constant == Theorems ==

• • • • • • • Euclid–Euler theorem, characterizing even perfect numbers • Euler's theorem, on modular exponentiation • Euler's partition theorem relating the product and series representations of the Euler function Π(1 − xn) • Goldbach–Euler theorem, stating that sum of 1/(k − 1), where k ranges over positive integers of the form mn for m ≥ 2 and n ≥ 2, equals 1 • == Laws ==

• Euler's first law, the sum of the external forces acting on a rigid body is equal to the rate of change of linear momentum of the body. • Euler's second law, the sum of the external moments about a point is equal to the rate of change of angular momentum about that point. == Other things ==

== Topics by field of study ==

Selected topics from above, grouped by subject, and additional topics from the fields of music and physical systems Analysis: derivatives, integrals, and logarithms Geometry and spatial arrangement Graph theory • Euler characteristic (formerly called Euler number) in algebraic topology and topological graph theory, and the corresponding Euler's formula \chi(S^2)=F-E+V=2 • Eulerian circuit, Euler cycle or Eulerian path – a path through a graph that takes each edge once • Eulerian graph has all its vertices spanned by an Eulerian path • Euler class • Euler diagram – popularly called "Venn diagrams", although some use this term only for a subclass of Euler diagrams. • Euler tour technique Music • Euler–Fokker genus • Euler's tritone Number theory • Euler's criterion – quadratic residues modulo by primes • Euler product – infinite product expansion, indexed by prime numbers of a Dirichlet series • Euler pseudoprime • Euler–Jacobi pseudoprime • Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer. • Euler system • Euler's factorization method Physical systems Polynomials • Euler's homogeneous function theorem, a theorem about homogeneous polynomials. • Euler polynomials • Euler spline – splines composed of arcs using Euler polynomials == See also ==