...the number $V$ of its vertices plus the number $F$ of its faces minus the number $E$ of its edges is equal to 2:
Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ t
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If an integer $a$ is not divisible by a prime number $p>2$, then the congruence
...be a [[quadratic residue]] or non-residue modulo $p$. It was proved by L. Euler in 1761 (see [[#References|[1]]]).
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...act that the set of [[prime number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic relation
[[Category:Number theory]]
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''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 totient function, Euler totient''
Another frequently used named for the [[Euler function]] $\phi(n)$, which counts a [[reduced system of residues]] modulo
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where $s$ is a real number and $p$ runs through all prime numbers. This product converges absolutely f
See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
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...n the above examples these are: 09 for $1/11$ and 0 or 9 for $7/4$. If the number is irrational, the infinite decimal fraction cannot be recurrent (e.g. $\sq
...$10^n-1$. Thus, the period length divides $\phi(q)$, the [[Euler function|Euler function]].
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$#C+1 = 37 : ~/encyclopedia/old_files/data/E036/E.0306400 Euler characteristic
is the number of $ k $-
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...Namely, he proved that for every number $a$ relatively prime to the given number $m>1$ there is the congruence
where $\phi(m)$ is the [[Euler function|Euler function]]. Another generalization of Fermat's little theorem is the equati
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The number $\gamma$ defined by
considered by L. Euler (1740). Its existence follows from the fact that the sequence
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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 $\sigma
The Euler identity can be generalized in the form
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''$r$-dimensional Betti number $p^r$ of a complex $K$''
...ant of the polyhedron which realizes the complex $K$, and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For i
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$#C+1 = 23 : ~/encyclopedia/old_files/data/E036/E.0306550 Euler polynomials
are the [[Euler numbers]]. The Euler polynomials can be computed successively by means of the formula
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...|coprime]] $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's [[totient function]], which is $J_1$.
* Ram Murty, M. Problems in Analytic Number Theory, ser. Graduate Texts in Mathematics '''206''' Springer-Verlag (2001)
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...artitions of an integer $n$ into parts equal to $a_1,\ldots,a_m$, i.e. the number of solutions in non-negative integers $x_1,\ldots,x_m$ of the equation
The simplest method of computing a denumerant is by Euler's recurrence relation:
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of the number $ n $,
is the number of integers $ k $
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3) The
number of terms in the Farey series of order $n$ is equal to
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Bukhshtab, "
Number theory" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[2]</T
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...ence $x^3\equiv a$ ($\bmod\,p$) may be checked for solvability using the [[Euler criterion]]: The congruence $x^3\equiv a$ ($\bmod\,p$), $(a,p)=1$, is solva
...modulo $p$. It follows from the criterion, in particular, that for a prime number $p$, the sequence of numbers $1,\dots,p-1$ contains exactly $(q-1)(p-1)/q$
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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
...D valign="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</TD></TR></tabl
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...ce the factorization into prime factors in $\mathbf Q(\sqrt d)$ of a prime number that does not divide $d$ depends on whether or not $x^2-d$ is reducible mod
...gn="top">[2]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translati
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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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...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.
...and quadruply-connected; the graph has no loops or multiple edges, and the number $ n $
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...mal [[Euler identity]] beween the Dirichlet series $L(a,s)$ and a formal [[Euler product]] over primes
and is [[Totally multiplicative function|totally multiplicative]] if the Euler product is of the form
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...length of a circle to its diameter; it is an infinite non-periodic decimal number
One frequently arrives at the number $\pi$ as the limit of certain arithmetic sequences involving simple laws. A
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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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...intersections some part of the $(n+1)$-dimensional space. Here $n$ is the number of unknown functions $y_i(x)$, $i=1,\dots,n$, on which the functional to be
depends. Euler's equation is understood in the vector sense, that is, it is a system of $n
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''number-theoretic function''
...er than $x$ — describes the distribution of primes; $\pi(x,q,l)$ gives the number of primes not larger than $x$ in the arithmetic progression $p\equiv l\pmod
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is the base of the natural logarithm, which is also known as the Napier number. This function is defined for any value of $ z $ (real or complex) by
i.e. for any natural number $ b > 0 $,
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''for the number of divisors''
...Obtained by P. Dirichlet in 1849; he noted that this sum is equal to the number of points $(x,y)$ with positive integer coordinates in the domain bounded b
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...number of all primitive roots of order $m$ is equal to the value of the [[Euler function]] $\phi(m)$ if $\mathrm{hcf}(m,\mathrm{char}(K)) = 1$.
In the field of [[complex number]]s, there are primitive roots of unity of every order: those of order $m$ t
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...ist, after which it is demonstrated that at least one of them is a complex number. C.F. Gauss was the first to prove the fundamental theorem of algebra witho
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$#C+1 = 34 : ~/encyclopedia/old_files/data/P074/P.0704530 Prime number
A natural number (positive [[Integer|integer]]) $ p > 1 $
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$#C+1 = 33 : ~/encyclopedia/old_files/data/L057/L.0507990 Lefschetz number
By definition, the Lefschetz number of $ f $
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is a natural number. If the congruence (1) is solvable, $ a $
...dulus power congruence problem there is a solvability criterion, due to L. Euler: For the congruence
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...apping|Degree of a mapping]]). It is related to the [[Euler characteristic|Euler characteristic]]. See also [[Poincaré theorem|Poincaré theorem]]; [[Krone
Cf. also [[Rotation number|Rotation number]] of a curve, which is the rotation of the unit tangent vector field of the
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...we have $\lambda(p^a) = \phi(p^a) = p^{a-1}(p-1)$ where $\phi$ denotes the Euler [[totient function]]. For powers of 2, we have $\lambda(2) = 1$, $\lambda(
A [[Carmichael number]] is a composite number with the property that $\lambda(n)$ divides $n-1$.
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''of a number $ a $
is a prime number; consequently, the notion of an index is only defined for these moduli.
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Let $f$, $g$ be functions on the [[natural number]]s. We say that $f$ has average order $g$ if the [[asymptotic equality]]
* The average order of $d(n)$, the [[number of divisors]] of $n$, is $\log n$;
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...nction $\sigma(m)$, the [[sum of divisors]] of a natural number $m$; the [[Euler function]] $\phi(m)$; and the [[Möbius function]] $\mu(m)$. The function $
Formally, the [[Dirichlet series]] of a multiplicative function $f$ has an [[Euler product]]:
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...problems of a much more general nature. The problem of writing every even number as a sum of two prime numbers has not yet (1989) been solved.
...oved that every sufficiently large even number is a sum of a prime and a number composed of at most two primes. For a proof see {{Cite|HaRi|Chapt. 11}}.
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Traditionally, a composite natural number $n$ is called a pseudo-prime if $2^{n-1} \equiv 1$ modulo $n$, for it has l
More recently, the concept has been extended to include any composite number that acts like a prime in some realization of a [[probabilistic primality t
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...field on a manifold|Vector field on a manifold]]) and let it have a finite number of isolated singular points $ A _ {1}, \dots, A _ {k} $.
is the [[Euler characteristic|Euler characteristic]] of $ V $.
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The formula expressing the rule for raising a [[Complex number|complex number]], expressed in trigonometric form
...power. According to de Moivre's formula the modulus $\rho$ of the complex number is raised to that power and the argument $\varphi$ is multiplied by the exp
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...or]] of two integers $a \ge b > 0$ is quite fast. It can be shown that the number of steps required is at most
...rs $(a,b)$ yields the [[continued fraction]] development of the [[rational number]] $a/b$.
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...mechanics)|Lagrange equations (in mechanics)]] (or to the [[Euler equation|Euler equation]] in the classical calculus of variations), in which the unknown m
is the number of degrees of freedom of the system, and he defined the function
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A field with a finite number of elements. First considered by E. Galois [[#References|[1]]].
...d|characteristic]] of this field. For any prime number $p$ and any natural number $n$ there exists a (unique up to an isomorphism) field of $p^n$ elements. I
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is the [[Euler constant|Euler constant]]. As a function of the complex variable $ z $,
the number of primes smaller than $ x $
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This function occurs in number theory as the limit
where $\Phi(x,y)$ denotes the number of positive integers not exceeding $x$ that are free of prime factors small
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An $n$-th root of a number $a$ is a number $x=a^{1/n}$ whose $n$-th power $x^n$ is equal to $a$.
...U_n$ is given by the [[Euler function|Euler function]] $\phi(n)$, i.e. the number of residues $\bmod\,n$ which are relatively prime to $n$. In a field of cha
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should pass are given. Since the general solution of the [[Euler equation|Euler equation]] of the simplest problem depends on two arbitrary constants, $
...ven manifolds. For instance, if in the [[Bolza problem|Bolza problem]] the number of boundary conditions to be satisfied by the sought curve $ x = ( x _ {1
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