...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
2 KB (304 words) - 19:26, 14 August 2014