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
3 KB (516 words) - 17:58, 8 November 2014
...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 $.
2 KB (308 words) - 06:03, 19 March 2022
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
1 KB (241 words) - 10:03, 4 June 2013
...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]] $a/b$.
2 KB (351 words) - 20:40, 16 November 2023
...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
3 KB (478 words) - 19:43, 5 June 2020
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
4 KB (749 words) - 18:32, 2 November 2014
is the [[Euler constant|Euler constant]]. As a function of the complex variable $ z $,
the number of primes smaller than $ x $
4 KB (515 words) - 18:28, 25 February 2021
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
2 KB (243 words) - 15:57, 22 September 2017
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
4 KB (680 words) - 13:40, 30 December 2018
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
4 KB (531 words) - 08:28, 6 June 2020
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