$#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 _
6 KB (877 words) - 19:38, 5 June 2020
...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 $
4 KB (632 words) - 20:10, 15 March 2023
...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
2 KB (358 words) - 17:25, 11 November 2023
...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
2 KB (352 words) - 14:43, 14 February 2020
$#C+1 = 61 : ~/encyclopedia/old_files/data/E036/E.0306620 Euler transformation
The Euler transformation of series. Given a series
6 KB (972 words) - 12:59, 6 January 2024
...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
2 KB (411 words) - 16:38, 24 November 2018
''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
4 KB (608 words) - 08:18, 4 November 2023
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 $,
6 KB (893 words) - 17:39, 20 January 2022
''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
951 bytes (154 words) - 08:27, 30 December 2015
...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
3 KB (496 words) - 07:46, 20 December 2014
...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
2 KB (348 words) - 05:59, 20 August 2014
$#C+1 = 34 : ~/encyclopedia/old_files/data/P074/P.0704530 Prime number
A natural number (positive [[Integer|integer]]) $ p > 1 $
6 KB (955 words) - 19:40, 1 November 2023
$#C+1 = 33 : ~/encyclopedia/old_files/data/L057/L.0507990 Lefschetz number
By definition, the Lefschetz number of $ f $
3 KB (494 words) - 08:31, 6 January 2024
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
5 KB (717 words) - 08:27, 6 June 2020
...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
2 KB (376 words) - 18:01, 4 November 2014
...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 $n-1$.
1 KB (186 words) - 16:57, 25 November 2023
''of a number $ a $
is a prime number; consequently, the notion of an index is only defined for these moduli.
5 KB (642 words) - 22:12, 5 June 2020
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 $\log n$;
1 KB (231 words) - 19:41, 17 November 2023
...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]]:
3 KB (419 words) - 20:15, 19 November 2017
...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}}.
3 KB (514 words) - 21:14, 9 January 2015