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 used trial division, improving on Cataldi's method, so that at most 372 divisions were needed.

colors are needed to color any graph embedded in a surface of Euler characteristic.

Now let us choose triangulations of S and S ′ with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics.

Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of, and were used by mathematicians including Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.

The Riemann zeta function or Euler – Riemann zeta function, ζ ( s ), is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s is greater than 1.

where, by definition, the left hand side is ζ ( s ) and the infinite product on the right hand side extends over all prime numbers p ( such expressions are called Euler products ):

Such infinite products are today called Euler products.

That is, the parameter is the curve length measured from the origin ( 0, 0 ) and the Euler spiral has infinite length.

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

This is now read as an ' extra ' factor in the Euler product for the zeta-function, corresponding to the infinite prime.

The first proof was found by Euler after much effort and is based on infinite descent.

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

* Project Euler a series of challenging mathematical / computer programming problems

In the context of computing asymptotic expansions of sums and series, usually the most useful form of the Euler – Maclaurin formula is

Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

Notably, Euler directly proved the power series expansions for and the inverse tangent function.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1.

The Meissel – Mertens constant is analogous to the Euler – Mascheroni constant, but the harmonic series sum in its definition is only over the primes rather than over all integers and the logarithm is taken twice, not just once.

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

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions.

The Fourier series is named in honour of Joseph Fourier ( 1768 – 1830 ), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d ' Alembert, and Daniel Bernoulli.

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

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.

Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.

In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function.

The Euler product attached to the Riemann zeta function, using also the sum of the geometric series, is

where ƒ is a smooth function, you could use the Euler – Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S ( x ), then for large values of a you could use " stationary phase " method to calculate the integral and give an approximate evaluation of the sum.

More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime p is the inverse of the Hecke polynomial, a quadratic polynomial in p < sup >− s </ sup >.

The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron.

Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle ( whence right triangles become meaningless ) and of equality of length of line segments in general ( whence circles become meaningless ) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments ( so line segments continue to have a midpoint ).

In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

If one bends and deforms the surface, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will.

They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736.

This can be explained intuitively by the fact that a quaternion describes a rotation in one single move (" please turn radians around the axis driven by vector "), while the Euler angles are made of three successive rotations.

In the language of stable homotopy theory, the Chern class, Stiefel-Whitney class, and Pontryagin class are stable, while the Euler class is unstable.

Euler rotations are a set of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant.

Usually the angle and axis pair is easier to work with, while the rotation vector is more compact, requiring only three numbers like Euler angles.

Euler rotations are defined as the movement obtained by changing one of the Euler angles while leaving the other two constant.

This base a is called an Euler witness for n ; it is a witness for the compositeness of n. The base a is called an Euler liar for n if the congruence is true while n is composite.

Observations show that the figure axis exhibits an annual wobble forced by surface mass displacement via atmospheric and / or ocean dynamics, while the free nutation is much larger than the Euler period and of the order of 435 to 445 sidereal days.