A formula which expresses the number $ \pi /2 $
Formula (1) is a direct consequence of Euler's product formula
3 KB (349 words) - 21:31, 29 December 2020
...$ counts the number of different partitions of $n$, so that $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [[#References|[a1]]]
...composition being distinct (see [[#References|[a3]]]). See also [[Additive number theory]]; [[Additive problems]].
2 KB (231 words) - 15:33, 11 November 2023
Any number $ a $,
Already in the Middle Ages a number of algorithms for constructing magic squares of odd order $ n $
5 KB (762 words) - 19:27, 12 January 2024
...the [[aliquot divisor]]s of the other, i.e. of the divisors other than the number itself. This definition is found already in Euclid's Elements and in the wo
...als of the amicable numbers converges {{Cite|Po1}}. If $A(x)$ denotes the number of integers $\le x$ that belong to an amicable pair, then it is known that
2 KB (281 words) - 14:01, 12 November 2023
[[Euler product|Euler product]] with characters. They are the basic instrument for studying by an
...$\rho : G \to {\textrm{GL}}(n,\C)$ of the Galois group $G$ of an algebraic number field $K$ (cf.
2 KB (347 words) - 21:23, 9 January 2015
...rational number $-r''$: $-r'>-r''$. The absolute value $|r|$ of a rational number $r$ is defined in the usual way: $|r|=r$ if $r\geq0$ and $|r|=-r$ if $r<0$.
...om its equivalence class, i.e. $a/b\in r$. Thus, one and the same rational number can be written as distinct, but equivalent, rational fractions.
6 KB (1,000 words) - 15:36, 14 February 2020
...ely from the formula given in [[Classical combinatorial problems]] for the
number of ''[[derangement]]s'': permutations $\pi$ such that $\pi(i)\neq i$ for al
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Knobloch, "
Euler and the history of a problem in probability theory" ''Ganita–Bharati'' ,
1 KB (177 words) - 07:34, 2 December 2016
$#C+1 = 22 : ~/encyclopedia/old_files/data/N067/N.0607940 Number theory
...trical figures, were the first and the most ancient mathematical concepts. Number theory arose from problems in [[Arithmetic|arithmetic]] connected with the
10 KB (1,503 words) - 08:03, 6 June 2020
can be summed by the Abel method ($A$-method) to the number $S$ if, for any real $x$, $0<x<1$, the series
This summation method can already be found in the works of L. Euler and G. Leibniz. The name "Abel summation method" originates from the [[Abel
2 KB (258 words) - 12:23, 10 January 2015
$#C+1 = 89 : ~/encyclopedia/old_files/data/E036/E.0306440 Euler equation
This equation was studied in detail by L. Euler, starting from 1740.
13 KB (1,798 words) - 19:06, 26 January 2022
...g concrete numbers $M_n$ it has been shown, for example, that $M_{31}$ (L. Euler, 1750) and $M_{61}$ (I.M. Pervushin, 1883) are Mersenne numbers. Computers
The exponents $n$ such that the Mersenne number $M_n$ is prime that were known before 1970 are
2 KB (344 words) - 06:44, 22 March 2024
$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306520 Euler\ANDMacLaurin formula
then the Euler–MacLaurin formula becomes
5 KB (745 words) - 09:18, 6 January 2024
...at is, a [[Complex number|complex number]] which is not real; an imaginary number of the form $iy$ is called purely imaginary (sometimes only these numbers a
Usually the phrases "imaginary number" and "complex number" just mean the same, "imaginary" being the historically chosen word and
3 KB (437 words) - 07:46, 4 November 2023
Inequalities following from Morse theory and relating the number of critical points (cf. [[Critical point|Critical point]]) of a Morse funct
...th $ n $-dimensional manifold $ M $ (without boundary) having a finite number of critical points. Then the [[Homology group|homology group]] $ H _ \la
6 KB (845 words) - 01:33, 5 March 2022
...domains (faces). Such a subdivision of the plane is known as a planar map. Euler's formula
...e $n$ is the number of vertices, $m$ is the number of edges and $r$ is the number of faces of the map (including the exterior face) applies to any planar map
5 KB (774 words) - 16:51, 16 March 2023
The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of
...d) have been an important driving force for the development of [[algebraic number theory]] and [[class field theory]]. A far-reaching generalization of the q
2 KB (295 words) - 17:43, 19 December 2014
..., the [[Reynolds number|Reynolds number]] and the [[Prandtl number|Prandtl number]].
...with the mean velocity $u$ divided by the viscosity $\nu$, and the Prandlt number is the ratio of the viscosity and the thermometric conductivity $\kappa$:
9 KB (1,338 words) - 15:30, 1 July 2020
is the Euler function.
is an algebraic number field, the only prime divisors that may be ramified in $ k ( \zeta _ {n}
3 KB (435 words) - 18:23, 2 January 2021
...those values of $\alpha$ that could not be well approximated by a rational number with a small denominator. Vinogradov's estimate used the sieve of Eratosthe
...}^\infty n^{-s}$ for complex numbers $s$ with real part exceeding $1$. The Euler product formula states that
10 KB (1,642 words) - 12:01, 13 February 2024
...he point $O(a,b,c)$ into which the origin is mapped and the [[Euler angles|Euler angles]], i.e. this group of motions is hexaparametric.
...ong to the segment of the natural number series $[1,n^2+n]$, but not every number of this segment can be an order of the complete group of motions.
6 KB (940 words) - 09:43, 11 August 2014