''Euler's totient function''
...ot exceeding $n$ and relatively prime to $n$ (the "totatives" of $n$). The Euler function is a [[multiplicative arithmetic function]], that is $\phi(1)=1$ a
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''Euler–Lagrange operator''
...for variational problems must satisfy (cf. also [[Euler–Lagrange equation|Euler–Lagrange equation]]).
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See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
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...oriented $n$-dimensional [[Manifold|manifold]] may be calculated from the Euler class of the [[Tangent bundle|tangent bundle]] [[#References|[a1]]], p. 348
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The Euler–Frobenius polynomials $p _ { m } ( x )$ of degree $m - 1 \geq 0$ are char
...invariance of this kind is to look for an appropriate space with which the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ are attached i
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is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
...hen its terms $a_n$ satisfy the condition $a_n=o((2q+1)^n)$, $q\geq0$. The Euler summation method can also be applied for analytic continuation beyond the d
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$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306520 Euler\ANDMacLaurin formula
then the Euler–MacLaurin formula becomes
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#REDIRECT [[Euler function]]
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...s or right-angled, or both right-angled and isosceles. The segments of the Euler line satisfy the relation
This line was first considered by L. Euler (1765).
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A generalization of the Goldbach–Euler problem (cf. [[Goldbach problem|Goldbach problem]]) according to which any
...of the distribution of prime numbers allows one to solve both the Goldbach–Euler problem and the more general problem of solvability of the linear Diophanti
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...(x,y)$ is zero. Particular integrals for it were obtained by G. Monge. The Euler–Lagrange equation was systematically investigated by S.N. Bernshtein, who
The Euler–Lagrange equation can be generalized with respect to the dimension: The e
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...3]]], the colliding of gravitational waves [[#References|[a6]]], etc.. The Euler–Poisson–Darboux equation has rather interesting properties, e.g. in rel
A formal solution to the Euler–Poisson–Darboux equation has the form [[#References|[a8]]]
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...ents satisfy certain compatibility conditions. Generally, almost all known Euler systems satisfy the condition ES) described below. Let $K$ be a number fiel
...pping from $K ( L l )$ down to $K ( L )$. Next to condition ES), any given Euler system may have additional properties, cf. [[#References|[a4]]], [[#Referen
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See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
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...that starts at the vertex. The centroid lies on the [[Euler straight line|Euler line]].
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...itrary real number and the product extends over all prime numbers $p$. The Euler identity also holds for all complex numbers $s = \sigma + it$ with $\sigma
The Euler identity can be generalized in the form
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considered by L. Euler (1740). Its existence follows from the fact that the sequence
...mber $\gamma$ is also known as the ''Euler-Mascheroni'' constant, after L. Euler (1707–1783) and L. Mascheroni (1750–1800).
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A smooth solution of the [[Euler equation|Euler equation]], which is a necessary extremum condition in the problem of varia
Euler's equation has the form
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$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306530 Euler method
by a polygonal line (Euler's polygonal line) whose segments are rectilinear on the intervals $ [ x _
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...ides the order of the group. Fermat's little theorem was generalized by L. Euler to the case modulo an arbitrary $m$. Namely, he proved that for every numbe
where $\phi(m)$ is the [[Euler function|Euler function]]. Another generalization of Fermat's little theorem is the equati
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..., 1740). There exists a constant $\gamma>0$, known as the [[Euler constant|Euler constant]], such that $S_n = \ln n + \gamma + \varepsilon_n$, where $\lim\l
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...$10^n-1$. Thus, the period length divides $\phi(q)$, the [[Euler function|Euler function]].
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is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
...hen its terms $a_n$ satisfy the condition $a_n=o((2q+1)^n)$, $q\geq0$. The Euler summation method can also be applied for analytic continuation beyond the d
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''$B$-function, Euler $B$-function, Euler integral of the first kind''
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is equal to its [[Euler characteristic|Euler characteristic]]. Betti numbers were introduced by E. Betti [[#References|[
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The simplest method of computing a denumerant is by Euler's recurrence relation:
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...a is employed in the calculus of variations to derive the [[Euler equation|Euler equation]] in its integral form. In this proof it is not necessary to assum
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The spiral of Cornu is sometimes called the spiral of Euler after L. Euler, who mentioned it first (1744). Beginning with the works of A. Cornu (1874)
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where $c=0.5772\ldots$ is the [[Euler constant|Euler constant]] and $\operatorname{Ci}(x)$ is the [[Integral cosine|integral cos
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$#C+1 = 61 : ~/encyclopedia/old_files/data/E036/E.0306620 Euler transformation
The Euler transformation of series. Given a series
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...the existence of Euler cycles (Euler's theorem): A connected graph has an Euler cycle if and only if each of its vertices (except two) has even degree.
A graph is said to be Hamiltonian (Eulerian) if it has a Hamilton (Euler) cycle. A graph is said to be Hamilton-connected if any two of its vertices
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...ence is unsolvable, then $a$ is called a quadratic non-residue modulo $m$. Euler's criterion: Let $p>2$ be prime. Then an integer $a$ coprime with $p$ is a
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...n the $(x,y)$-plane for which the general integral of the [[Euler equation|Euler equation]] can be represented in the form
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