The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

For closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics.

The Swiss mathematician Leonhard Euler pioneered the modern approach to congruence in about 1750, when he explicitly introduced the idea of congruence modulo a number N.

In number theory, the Euler numbers are a sequence E < sub > n </ sub > of integers defined by the following Taylor series expansion:

Alternatively, it is possible to show that any bridgeless bipartite planar graph with n vertices and m edges has by combining the Euler formula ( where f is the number of faces of a planar embedding ) with the observation that the number of faces is at most half the number of edges ( because each face has at least four edges and each edge belongs to exactly two faces ).

This means that as the number of contours increases, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.

The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere.

In number theory, Euler's theorem ( also known as the Fermat – Euler theorem or Euler's totient theorem ) states that if n and a are coprime positive integers, then

If the Euler criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol.

Orientation-free metrics of a group of connected or surrounded pixels include the Euler number, the perimeter, the area, the compactness, the area of holes, the minimum radius, the maximum radius.

In number theory, an odd composite integer n is called an Euler – Jacobi pseudoprime to base a, if a and n are coprime, and

Note that for orientable compact surfaces without boundary, the Euler characteristic equals, where is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and counts the number of handles.

More generally, if the polyhedron has Euler characteristic ( where g is the genus, meaning " number of holes "), then the sum of the defect is

The difference between the nth harmonic number and the natural logarithm of n converges to the Euler – Mascheroni constant.

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic ( or Euler – Poincaré characteristic ) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

where B < sub > k </ sub > is a Bernoulli number and R < sub > m, n </ sub > is the remainder term in the Euler – Maclaurin formula.

Bridge 8: Euler walks are possible if exactly zero or two nodes have an odd number of edges.

Amongst the fruits of his industry may be mentioned a laborious investigation of the disturbances of Jupiter by Saturn, the results of which were employed and confirmed by Euler in his prize essay of 1748 ; a series of lunar observations extending over fifty years ; some interesting researches in terrestrial magnetism and atmospheric electricity, in the latter of which he detected a regular diurnal period ; and the determination of the places of a great number of stars, including at least twelve separate observations of Uranus, between 1750 and its discovery as a planet.

M. E. Baron has noted that Leibniz ( 1646 – 1716 ) in the 17th century produced similar diagrams before Euler, but much of it was unpublished.

The Euler – Lagrange equations of motion for the functional E are then given in local coordinates by

This may also be written in terms of the Euler numbers E < sub > k </ sub > as

#( Functional equation and Poincaré duality ) The zeta function satisfies < dl >< dd ></ dl > or equivalently < dl >< dd ></ dl > where E is the Euler characteristic of X.

* The Euler product for the zeta function of E < sup > k </ sup > is

When k is even the coefficients of all its Euler factors are non-negative, so that each of the Euler factors has coefficients bounded by a constant times the coefficients of Z ( E < sup > k </ sup >, T ) and therefore converges on the same region and has no poles in this region.

* The eigenvalues of Frobenius on H ( U, E ) can now be estimated as they are the zeros of the zeta function of the sheaf E. This zeta function can be written as an Euler product of zeta functions of the stalks of E, and using the estimate for the eigenvalues on these stalks shows that this product converges for | T |< q < sup >− d / 2 − 1 / 2 </ sup >, so that there are no zeros of the zeta function in this region.

* Normalization: If E is a line bundle, then where is the Euler class of the underlying real vector bundle.

If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.

restricted to E. Then the analytical index of D is the holomorphic Euler characteristic of V:

where denotes the Euler class of E, and denotes the cup product of cohomology classes ; under the splitting principle, this corresponds to the square of the Vandermonde polynomial being the discriminant.

Substituting V and E into the Euler relation solved for F, one then obtains

An L-function L ( E, s ) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form.

* Robert E. Bradley, President of The Euler Society and chairman of the History of Mathematics Special Interest Group of the Mathematical Association of America.

His discovery of the 6th, that corresponding to p = 17 in the formula M < sub > p </ sub >= 2 < sup > p </ sup >- 1, exploded a many-times repeated number-theoretical myth ( Until Cataldi, 19 authors going back to Nicomachus are reported to have made the claim in L. E. Dickson's History of the Theory of Numbers — with a few more repeating this afterward ) that the perfect numbers had units digits that invariably alternated between 6 and 8 ; and that of the 7th ( for p = 19 ) held the record for the largest known prime for almost two centuries, until Leonhard Euler discovered that 2 < sup > 31 </ sup >-1 was the eighth Mersenne prime.

Melchior observed that, for any graph embedded in RP < sup > 2 </ sup >, the formula V − E + F must equal 1, the Euler characteristic of RP < sup > 2 </ sup >; where V, E, and F, are the number of vertices, edges, and faces of the graph, respectively.

After nearly two centuries, M < sub > 31 </ sub > was verified to be prime by Euler in 1772.

It was not until the 18th century that Leonhard Euler proved that the formula 2 < sup > p − 1 </ sup >( 2 < sup > p </ sup >− 1 ) will yield all the even perfect numbers.

Writing coordinates ( t, x ) = ( x < sup > 0 </ sup >, x < sup > 1 </ sup >, x < sup > 2 </ sup >, x < sup > 3 </ sup >) = x < sup > μ </ sup >, this form of the Euler – Lagrange equation is

Later, Nikolaus Bernoulli made the same assertion concerning the polynomial x < sup > 4 </ sup > − 4x < sup > 3 </ sup > + 2x < sup > 2 </ sup > + 4x + 4, but he got a letter from Euler in 1742 in which he was told that his polynomial happened to be equal to

Lambert first considered the related Lambert's Transcendental Equation in 1758, which led to a paper by Leonhard Euler in 1783 that discussed the special case of we < sup > w </ sup >.

If ( d < sub > n </ sub >: A < sub > n </ sub > → A < sub > n-1 </ sub >) is a chain complex such that all but finitely many A < sub > n </ sub > are zero, and the others are finitely generated abelian groups ( or finite dimensional vector spaces ), then we can define the Euler characteristic

Because the remainder R < sub > m, n </ sub > in the Euler – Maclaurin formula satisfies

An Euler probable prime to base a is an integer that is indicated prime by the somewhat stronger theorem that for any prime p, a < sup >( p − 1 )/ 2 </ sup > equals modulo p, where is the Legendre symbol.

The Dirichlet series generating function is especially useful when a < sub > n </ sub > is a multiplicative function, when it has an Euler product expression in terms of the function's Bell series

If a point P is chosen on the Euler line HN of the reference triangle ABC with a position vector < u > p </ u > such that < u > p </ u >