# REDIRECT Euler's conjecture

** Roger Frye used experimental mathematics techniques to find the smallest counterexample to Euler's sum of powers conjecture.

This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture.

He pursued mathematics as an amateur, his most famous achievement being his confirmation in 1901 of Leonhard Euler's conjecture that no 6 × 6 Graeco-Latin square was possible .< ref >

In 1988, he found a counterexample to Euler's sum of powers conjecture for fourth powers.

Ray-Chaudhuri ) and constructed ( with S. S. Shrikhande and E. T. Parker ) a Graeco-Latin square of size 10, a counterexample to Euler's conjecture that no Graeco-Latin square of size 4k + 2 exists.

Both Fermat's assertion and Euler's conjecture were established by Lagrange.

Schanuel's conjecture, if proved, would also settle the algebraic nature of numbers such as e + π and e < sup > e </ sup >, and prove that e and π are algebraically independent simply by setting z < sub > 1 </ sub > = 1 and z < sub > 2 </ sub > = πi, and using Euler's identity.

Euler's rule is a generalization of the Thâbit ibn Kurrah rule.

When a body is acted upon by external contact forces, internal contact forces are then transmitted from point to point inside the body to balance their action, according to Newton's second law of motion of conservation of linear momentum and angular momentum ( for continuous bodies these laws are called the Euler's equations of motion ).

The number of integers coprime to a positive integer n, between 1 and n, is given by Euler's totient function ( or Euler's phi function ) φ ( n ).

However, the phase of this contribution at any given point along the path is determined by the action along the path ( see Euler's formula ):

: This article is about Euler's formula in complex analysis.

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function.

This complex exponential function is sometimes denoted The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.

In fact, the same proof shows that Euler's formula is even valid for all complex numbers z.

Euler's identity is an easy consequence of Euler's formula.

Here is a proof of Euler's formula using power series expansions

One way to find that analytic continuation is to use Euler's integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,

Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy and L ' Huillier, and is at the origin of topology.

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: " Read Euler, read Euler, he is the master of us all.

Euler's name is associated with a large number of topics.

Euler is the only mathematician to have two numbers named after him: the immensely important Euler's Number in calculus, e, approximately equal to 2. 71828, and the Euler-Mascheroni Constant γ ( gamma ) sometimes referred to as just " Euler's constant ", approximately equal to 0. 57721.

This is known as Euler's first law.

Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.

e is Euler's Number, a fundamental mathematical constant.

# Compute, where φ is Euler's totient function.

The physicist Richard Feynman called Euler's formula " our jewel " and " one of the most remarkable, almost astounding, formulas in all of mathematics.

This work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler's personality and religious beliefs.

In the history of mathematics, Euler's solution of the Königsberg bridge problem is considered to be the first theorem of graph theory, a subject now generally regarded as a branch of combinatorics.

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

In mathematics, the series representation of Euler's number e

In mathematics, a Cauchy – Euler equation ( also known as the Euler – Cauchy equation, or simply Euler's equation ) is a linear homogeneous ordinary differential equation with variable coefficients.

( The trigonometric functions are in fact closely related to and can be defined via the exponential function using Euler's formula ).

which can be related to the usual sine and cosine forms using Euler's formula.

By Euler's formula the number of faces (' F '), vertices ( V ), and edges ( E ) of any convex polyhedron are related by the formula " F + V-E "

Euler's four-square identity is an analogous identity involving four squares instead of two that is related to quaternions.

and is related to the gamma function by Euler's reflection formula:

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem.

Fermat's little theorem is generalized by Euler's theorem: for any modulus n and any integer a coprime to n, we have

His first original research, comprising part of a proof of Fermat's last theorem for the case, brought him immediate fame, being the first advance in the theorem since Fermat's own proof of the case and Euler's proof for.