...fined on the set of all integers $t=0,\pm1,\pm2,\dots,$ or on the positive integers $t=1,2,\dots$.
305 bytes (44 words) - 07:32, 23 August 2014
The ring of Gaussian integers or [[Gauss number]]s, $\mathbb{Z}[i]$.
100 bytes (17 words) - 07:32, 28 November 2014
...of (ordered) $n$-tuples of integers (or non-negative integers or positive integers) for which there exists a polynomial $P(a_1,\ldots,a_n,z_1,\ldots,z_k)$ wit
1 KB (165 words) - 16:59, 14 February 2020
...ained as $x=a^2-b^2$, $y=2ab$, $z=a^2+b^2$, where $a$ and $b$ are positive integers $(a>b)$. The Pythagorean numbers can be interpreted as the sides of a right
761 bytes (116 words) - 07:30, 10 December 2016
...argument that satisfies the following conditions for two relatively prime integers $m,n$
...ndition $f(mn) = f(m) + f(n)$ is also satisfied for relatively non-coprime integers $m,n$ as well; in such a case $f(p^a) = a f(p)$.
1 KB (206 words) - 05:52, 15 April 2023
An integer is an element of the ring of integers $\mathbf Z=\{\dots,-1,0,1,\dots\}$. The ring $\mathbf Z$ is the minimal rin
...d of rational numbers, the [[field of fractions]] of $\mathbf Z$, then the integers of $k$ are the elements of the [[integral closure]] of $\mathbf Z$ in $k$.
2 KB (283 words) - 17:19, 30 November 2014
divisibility of integers by a given prime number $p$. The extension is
addition and the multiplication of $p$-adic integers is defined by the
6 KB (1,089 words) - 20:29, 9 April 2017
...sequence of all natural numbers belongs to a given sequence $A=\{a_k\}$ of
integers $a_0=0<1\leq a_1<\dotsb<a_k$. By the density of a sequence $A$ one means th
...f and only if $A$ coincides with the set $\mathbf N_0$ of all non-negative
integers. Let $A+B$ be the arithmetic sum of two sequences $A=\{a_k\}$ and $B=\{b_t\
3 KB (461 words) - 11:41, 14 February 2020
...weak Ditters conjecture, which states that $\mathcal{M}$ is free over the integers without giving a concrete set of generators, has been proved; see [[Quasi-s
1 KB (182 words) - 19:16, 17 June 2016
of positive integers not exceeding $ x $
...[a1]</td> <td valign="top"> K. Alladi, "The Turán–Kubilius inequality for integers without large prime factors" ''J. Reine Angew. Math.'' , '''335''' (1982)
2 KB (376 words) - 08:55, 10 November 2023
tuples of integers (non-negative integers, positive integers) for which it is possible to write down a Diophantine equation (cf. [[Dioph
...ermissible values of which are integers (non-negative integers or positive integers, respectively), and which is solvable for $ x _ {1} \dots x _ {l} $
7 KB (998 words) - 19:35, 5 June 2020
Let $S=\{a_1,\ldots,a_k\}$ be a finite set of positive integers with greatest common divisor $1$. The Frobenius number of $S$ is the larges
314 bytes (50 words) - 15:24, 10 August 2014
...positive integer $\lambda$ such that $a^\lambda \equiv 1 \pmod n$ for all integers $a$ coprime to $n$. It is equal to the [[least common multiple]] of its va
1 KB (186 words) - 16:57, 25 November 2023
...r programming]]) but for which some of the variables are constrained to be integers.
299 bytes (39 words) - 16:56, 7 February 2011
ranges over the positive integers (cf. [[Prime number|Prime number]]). There are some obvious necessary condi
n ^ {p} - n - p \equiv0 ( { \mathop{\rm mod} } p ) \textrm{ for all integers } n.
3 KB (382 words) - 06:29, 30 May 2020
...og\log n$ in the sense that, given any $\epsilon > 0$, almost all positive integers $n$ satisfy
...$\psi(n)$ tending to infinity as $n\rightarrow\infty$, almost all positive integers $n$ satisfy
4 KB (647 words) - 07:30, 18 March 2023
A generalization of the concept of divisibility of integers without remainder (cf. [[Division]]).
...ey are known as irreducible polynomials. Rings in which — like in rings of integers or polynomials — there is unique decomposition into prime factors (up to
2 KB (396 words) - 18:39, 25 September 2017
...\mathbb{Z}$ can be considered as a [[Small category|small category]] with integers as objects and all possible pairs $(i,j)$, where $i,j \in \mathbb{Z}$ and $
...icient to indicate a family of objects from $\mathfrak{K}$, indexed by the integers, and for each integer $i$ to choose a morphism $\alpha_{i,i+1} : A_i \right
2 KB (380 words) - 11:48, 26 October 2014
A method for finding the [[greatest common divisor]] of two integers, two polynomials (and, in general, two elements of a [[Euclidean ring]]) or
For two positive integers $a \ge b$, the method is as follows. Division with remainder of $a$ by $b$
2 KB (351 words) - 20:40, 16 November 2023
...recursive sequence]] of integers defined by two integer parameters. Given integers $P$, $Q$ with $D =P^2 - 4Q \neq 0$, the Lucas sequences of the first kind,
...\alpha, \beta$ algebraic numbers with $\alpha\beta$ and $(\alpha+\beta)^2$ integers:
2 KB (351 words) - 20:26, 20 November 2023
...of algebras: for example, the ring of integers modulo $2$ and the ring of integers modulo $3$ have no free product in the variety of rings with 1. However, co
2 KB (259 words) - 20:56, 30 July 2014
...the lengths of the sides and the surface area of which are expressible by integers. Named after Heron (1st century A.D.), who studied triangles with side leng
382 bytes (53 words) - 12:08, 13 August 2014
...al linear transference theorems concern the relations between solutions in integers of a system of homogeneous linear inequalities having a non-singular square
2 KB (255 words) - 20:56, 25 October 2014
A function defined on the set of positive integers whose range is contained in the set considered.
is the set of positive integers and $ X $
3 KB (560 words) - 08:13, 6 June 2020
...he above example makes it possible to consider the set of all non-negative integers — the [[Natural sequence|natural sequence]] — as a mathematical object.
2 KB (297 words) - 17:03, 7 February 2011
...is always possible, and the result of a division is unique. In the ring of integers division is not always possible. Thus, 10 is divisible by 5, but is not div
...e integers, then division with remainder of $a$ by $b$ consists of finding integers $x$ and $y$ such that
3 KB (464 words) - 18:40, 30 December 2018
...ces|[1]]]. Let $A = (0 < a_1 < a_2 < \cdots)$ be an increasing sequence of integers and let
...roof of the fundamental theorem on the density of sums of sets of positive integers" ''Ann. of Math.'' , '''43''' (1942) pp. 523–527 {{ZBL|0061.07406}}</
2 KB (286 words) - 11:41, 19 November 2017
Suppose now that the equation $A + B + C = 0$ holds for coprime integers $A,B,C$. The conjecture asserts that for every $\epsilon > 0$ there exist
2 KB (362 words) - 19:28, 14 November 2023
Problems in number theory concerning the decomposition (or partition) of integers into summands of a given kind. The solution of classical additive problems
3) The problem on the representation of positive integers as the sum of a bounded number of prime numbers (the weak Goldbach problem)
4 KB (528 words) - 17:45, 4 December 2014
...a]]: that is, exponentiation $x \mapsto x^n$ is well-defined for positive integers $n$, and $x^{m+n} = x^m \star x^n$. The set of powers of $x$ thus forms a
524 bytes (78 words) - 10:28, 1 January 2016
...only interested to represent in such a way all sufficiently large positive integers and speaks then of an asymptotic additive basis. For example, the set of sq
...for suitable constants $C_1$ and $C_2$, for all but finitely many positive integers $x$. See also [[#References|[a5]]] for a modified probabilistic constructio
4 KB (658 words) - 19:37, 29 March 2024
...ophantine equations]], for which the problem posed is to find solutions in integers, which can at the same time be considered as [[Additive problems|additive p
...terms of a desired type. Such problems include, for example, solutions in integers of the following equations:
2 KB (277 words) - 19:35, 5 June 2020
...sidue]]s. He also discovered the properties of the set $\Gamma$ of complex integers.
2 KB (278 words) - 20:01, 21 March 2023
...divisors of $b$ (possibly empty) is contained in $S$ are the so-called $S$-integers (corresponding to the specific set $S$). Clearly, this is a subring $R_S$ o
...$ containing all Archimedean valuations of $K$. Then, the set $R_S$ of $S$-integers and the set $R _ { S } ^ { * }$ of $S$-units are defined exactly as in the
5 KB (751 words) - 13:28, 25 November 2023
...or all $x$ of some predicate $P(x)$ defined on the set of all non-negative integers, if the following two conditions hold: 1) $P(0)$ is valid; and 2) for any $
...nstead of the induction axiom: Let $P(x)$ be some property of non-negative integers; if for any $x$ it follows from the assumption that $P(y)$ is true for all
2 KB (375 words) - 17:00, 30 December 2018
and a subsequence of integers $n_1 < n_2 < \cdots$, the distribution functions of the random variables
1 KB (181 words) - 20:38, 8 November 2017
...the language of arithmetic contains one type, namely for the non-negative integers.
597 bytes (87 words) - 17:21, 7 February 2011
Any set of $m$ integers that are incongruent $\bmod\,m$. Usually, as a complete residue system $\bm
428 bytes (69 words) - 12:43, 23 November 2014
...e unique integral object in the category of Abelian groups is the group of integers.
602 bytes (94 words) - 16:55, 7 February 2011
792 bytes (117 words) - 18:34, 11 April 2023
...$p$ is [[Isomorphism|isomorphic]] to the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p$, often denoted $\mathbb{F}_p$ or $\mathrm{GF}(p)$.
658 bytes (95 words) - 19:39, 27 October 2023
The set of positive integers $ a _ {0} $
of all positive integers $ a _ {0} $
4 KB (575 words) - 08:24, 6 June 2020
...smallest positive number among the common [[multiple]]s of a finite set of integers or, in particular, of [[natural number]]s, $a_1,\ldots,a_n$. The least comm
3) if the integers $a_1,\ldots,a_n$ are expressed as
2 KB (334 words) - 20:19, 2 November 2016
runs through all integers; some of the submodules $ A _ {n} $
...arbitrary groups may be introduced in a similar way. The gradations by the integers play an important role in the theory of projective algebraic varieties or s
2 KB (330 words) - 16:25, 23 November 2023
ranges over the positive integers. An obvious necessary condition is that all coefficients of $ f $
for the positive integers $ n $(
3 KB (469 words) - 10:15, 29 May 2020
* P.J. Cameron and P. Erdős, ''On the number of sets of integers with various properties'', Number theory (Banff, 1988), de Gruyter, Berlin
558 bytes (83 words) - 14:02, 30 December 2015
...n satisfying some arithmetic condition (usually one looks for solutions in integers or rational numbers). The study of such equations forms the topic of the th
608 bytes (91 words) - 17:19, 7 February 2011
Integers without common (prime) divisors. The [[greatest common divisor]] of two cop
667 bytes (98 words) - 18:57, 18 October 2014
...lopediaofmath.org/legacyimages/b/b015/b015220/b01522010.png" /> of natural
integers, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o
5 KB (703 words) - 17:01, 7 February 2011
...of many small prime numbers. Let $\Psi(x,y)$ denote the number of positive integers less than or equal to $x$ and free of prime divisors greater than $y$. When
...le \exp(\log^{(3/5)-\epsilon} y)$. There are similar results for algebraic integers, [[#References|[a3]]].
9 KB (1,347 words) - 05:11, 15 February 2024
* Kishore, Masao. "Odd integers N with five distinct prime factors for which $2−10^{−12} < \sigma(N)/N
...Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd), ''Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006''. CRM Pro
2 KB (260 words) - 19:44, 17 November 2023
...mial are rational integers. For instance, $i$ and $1+\sqrt2$ are algebraic integers, being roots of the polynomials $x^2+1$ and $x^2-2x-1$.
...c integers form an everywhere-dense set in $\mathbf R$, while the rational integers form a discrete set.
10 KB (1,645 words) - 17:08, 14 February 2020
...Z$ of integers, or the additive groups $\mathbf Z_n$ of residue classes of integers modulo $n$, $n=1,2,\dots$).
2 KB (343 words) - 18:24, 26 October 2014
where the minimum is taken over all possible sets of integers $a_1,\ldots,a_n$, not all zero, satisfying the conditions
724 bytes (124 words) - 19:48, 20 November 2014
are integers and if $ k \neq 0 $
If there exists integers $ n $
2 KB (331 words) - 17:32, 5 June 2020
...function $S(A;P,z)$, which denotes the number of elements of a set $A$ of integers that are not divisible by prime numbers $p < z$ and that belong to a certai
2 KB (268 words) - 19:56, 21 November 2014
...omorph of a group]]) is isomorphic to a group of matrices over the ring of integers; this enables one to use methods from algebraic geometry, number theory and
Every solvable linear group over the integers is polycyclic, [[#References|[a1]]]. A solvable group is polycyclic if and
4 KB (588 words) - 19:59, 11 April 2014
...] of [[countably infinite|countably]] many copies of the additive group of integers $\mathbb{Z}$. Reinhold Baer proved in 1937 that $\mathcal{B}$ is not [[Fre
775 bytes (113 words) - 07:30, 24 November 2023
...rly any field of characteristic zero is a binomial ring, as is the ring of integers $\mathbf{Z}$.
663 bytes (103 words) - 20:19, 20 November 2023
is the group of integers; and 4) for any right (correspondingly, left) ideal $ J $
...; [[Free module|Free module]]). The class of flat modules over the ring of integers coincides with the class of Abelian groups without torsion. All modules ove
2 KB (372 words) - 19:39, 5 June 2020
...eal numbers and let $N$ and $\epsilon$ be positive numbers; then there are integers $m$ and $p_1,\ldots,p_n$ such that
2 KB (349 words) - 12:10, 13 March 2018
...he semi-group $D$ of divisors (cf. [[Divisor|Divisor]]) of the ring $A$ of integers of an algebraic number field. The semi-group $D$ is a free commutative semi
...e absence of uniqueness of factorization into prime factors in the ring of integers of an algebraic number field. For every $a\in A$, the factorization of the
5 KB (853 words) - 12:16, 22 August 2014
...f pairs. A pair $(a,b)$ is also called a rational fraction (or fraction of integers). Distinct classes define distinct rational numbers. The set of all rationa
...orphism from the set of such rational numbers onto the ring $\mathbf Z$ of integers. Therefore, the rational number containing a rational fraction of the form
6 KB (1,000 words) - 15:36, 14 February 2020
...an irrational number, $c$ is an integer, and the unknowns $x$ and $y$ are integers.
...[2]</TD> <TD valign="top"> A.D. Gel'fond, "The solution of equations in integers" , Noordhoff (1960) (Translated from Russian)</TD></TR><TR><TD valign="to
2 KB (360 words) - 09:21, 27 April 2014
is the ring of integers of $ K $.
be a prime ideal of (the ring of integers $ A _ {K} $)
3 KB (461 words) - 22:12, 5 June 2020
are coprime integers and $ n \geq 2 $
...was forced to extend the ring of rational integers to the ring of complex integers $ \mathbf Z [ i] $.
5 KB (717 words) - 08:27, 6 June 2020
...by letters of the Slavic alphabet with a sign (titlo) written above them. Integers up to 999 were compiled by placing Slavic numerals in adjacent positions. T
1,012 bytes (155 words) - 17:18, 7 February 2011
...er]]s and $\mathbf{Z}_p$ is the additive group of the ring of all $p$-adic integers. A quasi-cyclic group is the union of an ascending chain of cyclic groups $
...phisms of a group of type $p^\infty$ is isomorphic to the ring of $p$-adic integers. A quasi-cyclic group coincides with its [[Frattini-subgroup(2)|Frattini su
3 KB (414 words) - 20:17, 16 October 2017
...ite cyclic group, which is isomorphic to the additive group $\mathbf Z$ of integers. A finite cyclic group $G$ of order $n$ is isomorphic to the additive group
666 bytes (123 words) - 06:43, 21 March 2024
...P^{v_{nm}}$ projective spaces over an arbitrary field (or over the ring of integers), regarded as schemes; let $u_0, \ldots, u_n$ be projective coordinates in
2 KB (408 words) - 23:35, 22 October 2018
...imates are obtained for the height of the solutions, either in the ring of integers of a fixed algebraic number [[Field|field]] $ K $,
integers, where $ S $
7 KB (1,087 words) - 19:41, 5 June 2020
...[a2]</TD> <TD valign="top"> P. Erdös, P. Turán, "On some sequences of integers" ''J. London Math. Soc.'' , '''11''' (1936) pp. 261–264</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Szemerédi, "On sets of integers containing no $k$-elements in arithmetic progression" ''Acta Arithm.'' , '
3 KB (415 words) - 19:05, 20 November 2023
...s of a unit. It is denoted by the symbol $a/b$, where $a$ and $b\ne 0$ are integers (cf.
Fractions may be represented as ordered pairs of integers $(a,b)$, $b\ne 0$, for which an equivalence relation has been specified (an
4 KB (649 words) - 17:15, 9 December 2013
...alisation of the [[Jacobi symbol]] $\left(\frac{a}{b}\right)$ to arbitrary integers $a$, $b$. If $b=0$, it is defined as 1 if $a = \pm 1$ and 0 otherwise. F
857 bytes (137 words) - 16:43, 23 November 2023
given on this space and taking values in the positive integers). The term "integral automorphism" is mostly used in the non-Soviet liter
1 KB (212 words) - 22:12, 5 June 2020
are integers are also periods of $ f ( z) $.
are integers. Analytic functions of one complex variable with more than two primitive pe
4 KB (565 words) - 19:36, 5 June 2020
...ma_0,\Gamma_1,\ldots\}$ of $X$ and a sequence $n_0,n_1,\ldots$ of positive integers greater than $N$ such that $\rho(f(x),f_{n_k}(x))<\epsilon$ for every $x\in
1,012 bytes (146 words) - 19:31, 9 November 2014
...er's conjecture holds for non-reciprocal integers $\alpha$, i.e. algebraic integers whose minimal polynomials do not have $1/\alpha$ as a root. In this case th
...by and H.L. Montgomery [[#References|[a1]]] established, for all algebraic integers $\alpha\ne0$ of degree $N$ that are not roots of unity, the inequality
7 KB (1,029 words) - 07:50, 27 March 2018
...tal numbers to prove that a system of linear inequalities can be solved in integers (cf. [[Dirichlet theorem|Dirichlet theorem]] in the theory of Diophantine a
928 bytes (141 words) - 19:35, 5 June 2020
...les of Gauss semi-groups include the multiplicative semi-group of non-zero integers, and that of non-zero polynomials in one unknown over a field. Any two elem
1 KB (153 words) - 16:17, 21 December 2014
...ediaofmath.org/legacyimages/h/h110/h110290/h1102907.png" /> be its ring of
integers (cf. also [[Extension of a field|Extension of a field]]; [[Norm on a field|
...f Tate–Oort Galois extensions can also be used to characterize the ring of
integers of certain degree-<img align="absmiddle" border="0" src="https://www.encycl
13 KB (1,801 words) - 19:17, 12 January 2018
are not integers, then equation (1) has a fundamental system of solutions of the form
are not integers, where the $ \lambda _ {j} $
3 KB (441 words) - 17:32, 5 June 2020
...dices $k=(k_1,\dots,k_m)$, $|k|=k_1+\dots+k_m$, and $k_i$ are non-negative integers. A MacLaurin series is a special case of a [[Taylor series|Taylor series]].
883 bytes (150 words) - 06:08, 13 June 2022
...ample, for a finite Abelian group$M$ regarded as a module over the ring of integers, $\mathrm{Supp}(M)$ consists of all prime ideals $(p)$, where $p$ divides t
918 bytes (149 words) - 20:46, 18 October 2014
...hi$ with the above-mentioned magma as its base. When $\Phi$ is the ring of integers, a free algebra over $\Phi$ is called a free ring (cf. [[Free associative a
1 KB (163 words) - 08:20, 12 November 2023
...ral numbers and hence of the same [[cardinality]]. For example, the set of integers, the set of [[rational number]]s or the set of [[algebraic number]]s.
908 bytes (146 words) - 08:50, 26 November 2023
in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $
are integers (Dantzig's theorem on integral solutions of the transport problem).
3 KB (453 words) - 18:48, 5 April 2020
...on a skew-field the group of values of which is isomorphic to the group of integers $ \mathbf Z $.
1 KB (165 words) - 19:36, 5 June 2020
A statement about the number of ways that positive integers can be represented as a sum of two elements of a given set, stating that th
990 bytes (160 words) - 16:49, 23 November 2023
are integers.
The idea of lattices and lattice points links geometry to arithmetic (integers). Therefore it plays a central role in the [[Geometry of numbers|geometry o
3 KB (421 words) - 22:15, 5 June 2020
...orics)]], an expression of a natural numbers as an ordered sum of positive integers; [[Composition series]], a maximal linearly ordered subset of a partially o
810 bytes (132 words) - 14:29, 3 September 2017
...y. For example, in elementary arithmetic a universal set is the set of all integers. The concept of a universal set plays a basic role in [[Set theory|set theo
1 KB (194 words) - 17:09, 7 February 2011
...gers for $j=1,\dots,p$, $p\leq n$, the $a_{ij}$, $b_i$ and $c_j$ are given integers, and the $x_j$ are variables.
where $t$, $x_1$, $x_2$, and $x_3$ are integers. If by some method of integer programming the answer obtained is a positive
5 KB (700 words) - 19:16, 4 November 2014
...[Arithmetic|arithmetic]] connected with the multiplication and division of integers.
...s (cf. [[Perfect number|Perfect number]]) was studied; and the solution in integers of the equation $ x ^ {2} + y ^ {2} = z ^ {2} $
10 KB (1,503 words) - 08:03, 6 June 2020
...urable with the group $G_O$ of $O$-points of $G$, where $O$ is the ring of integers of $k$. An arithmetic group $H \cap G_{\mathbb{R}}$ is a [[Discrete subgrou
4 KB (527 words) - 20:14, 14 October 2014
The function defined on the set of non-negative integers with value at $n$ equal to the product of the natural numbers from 1 to $n$
1 KB (162 words) - 13:41, 14 February 2020
...ref{eq1}, restricted to $[-1,1]$, are called [[Legendre polynomials]]; for integers $\mu$ with $-\nu \leq \mu \leq \nu$, the solutions of \ref{eq1}, restricted
956 bytes (140 words) - 20:14, 1 November 2014
...ts 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
1 KB (185 words) - 16:46, 23 November 2023
where $N$ is a system of sets of positive integers, called the base of the $\delta$-$\sigma$-operation. See [[Descriptive set
1 KB (203 words) - 17:25, 1 January 2021
are integers, the binary quadratic form is said to be integral. The expression $ d = a
can be represented as the sum of two squares of integers. The theory of quadratic forms was completed by J.L. Lagrange and by C.F. G
7 KB (977 words) - 15:17, 31 March 2024
...iple, be indefinitely extended (e.g. the successive generation of positive integers starting from zero), the abstraction of potential realizability consists in
1 KB (196 words) - 17:15, 7 February 2011
for all integers $m \ne 0$. Demonstrated in 1916 by H. Weyl. See [[Weyl method]].
1 KB (179 words) - 22:16, 12 March 2018
are relatively prime integers and $ g $
there exist integers $ \gamma , \gamma _ {0} \dots \gamma _ {s} $
5 KB (642 words) - 22:12, 5 June 2020
...a natural number $n$ is an expression of $n$ as an ordered sum of positive integers. Thus the compositions of $4$ are $4, 3+1, 1+3, 2+2, 2+1+1, 2+1+1, 1+1+2,
1 KB (143 words) - 14:20, 12 November 2023
...is ''irreducible''. For polynomials, see [[Irreducible polynomial]]; for integers, the traditional terminology is [[prime number]].
1 KB (209 words) - 08:06, 26 November 2023
of integers, the ring of integers of some algebraic number field, the field $ \mathbf R $
4 KB (512 words) - 10:59, 29 May 2020
Let $K$ be an algebraic number field with ring of integers $R$. A ''modulus'' is a formal product
...For a finite place $\mathfrak{p}$, that is, a prime ideal of the ring of integers, we define $x$ and $y$ to be congruent modulo $\mathfrak{p}^n$ if ''x''/''
3 KB (527 words) - 17:00, 23 November 2023
...s $\Vert \alpha q \Vert < \phi(q)$ have an infinite number of solutions in integers $q \ge 1$ for almost-all real numbers $\alpha$ if the series
where $\min$ is taken over all integers $n \in \mathbf{Z}$; the term "almost-all" refers to Lebesgue measure in t
8 KB (1,172 words) - 17:12, 8 March 2018
...of these order types coincides with the definition of the sum of positive integers. The following identities are valid: $$ (\alpha + \beta) + \gamma = \alpha
6 KB (1,008 words) - 17:31, 5 January 2017
Let $K$ be a set of positive integers and let $\lambda$ be a positive integer. A pairwise balanced design ($\oper
Let $K$ be a set of positive integers. Define $\alpha(K) = \gcd\{k-1: k \in K\}$ and $\beta(K) = \gcd \{k(k-1) :
8 KB (1,452 words) - 12:17, 13 February 2024
...] $n$, named after [[Camille Jordan]], counting the $k$-tuples of positive integers all less than or equal to $n$ that form a [[Coprime numbers|coprime]] $(k +
1 KB (172 words) - 13:07, 19 March 2023
...$ for which there exists an element $\alpha \in K$ such that the [[ring of integers]] $O_K$ is a polynomial ring $\mathbb{Z}[\alpha]$. The powers of such a el
1 KB (180 words) - 16:57, 25 November 2023
has infinitely many solutions in coprime integers $p,q$. The Lagrange spectrum $L$ is the set of all values taken by the fun
1 KB (191 words) - 16:51, 23 November 2023
adic integers for any prime $ p $,
of integers in its $ \mathbf Z $-
4 KB (561 words) - 17:31, 5 June 2020
...gebraic number]]) and let $\widetilde{\bf Z}$ be the ring of all algebraic integers. Then $\tilde {\bf Q }$ is the algebraic closure of $\mathbf{Q}$ and $\wide
...of algebraic integers|Local-global principles for large rings of algebraic integers]].
11 KB (1,771 words) - 16:57, 1 July 2020
...finding solutions of systems of such Diophantine equations in the ring of integers of any number field of finite degree over $ \mathbf Q $.
...lynomial equations having integral coefficients. The ring of all algebraic integers, $ {\widetilde{\mathbf Z} } $,
12 KB (1,775 words) - 11:58, 4 April 2020
...Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd). ''Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006''. CRM Pro
1 KB (164 words) - 07:50, 4 November 2023
th component. If the indices range over non-negative integers $ < m $,
...sequence). Much more important, if the indices range over all non-negative integers, this proves that there are uncountably many infinite sequences of zeros an
6 KB (881 words) - 20:32, 21 January 2021
...additive, and the ones usually considered are partition problems of large integers.
...d in 1748 by L. Euler, who employed power series to study the partition of integers into positive summands; in particular, he investigated the problem of parti
10 KB (1,609 words) - 13:03, 8 February 2020
...au(N)$ divides the [[sum of divisors]] $\sigma(N)$. The natural density of integers $N$ for which $d(N)^2$ divides $\sigma(N)$ is 1/2.
1 KB (172 words) - 09:52, 12 October 2023
...gebraic number fields are Dedekind, i.e. the integral closures of rings of integers in finite algebraic extensions of the field of rational numbers.
4 KB (637 words) - 17:32, 5 June 2020
The largest of the common divisors of a set of integers or, in particular, of natural numbers $a_1,\ldots,a_n$. The greatest common
...l ring]]). Examples of such domains are the ring $\mathbb{Z}$ of rational integers or polynomial rings $F[X]$, where $F$ is a [[field]] (e.g., $\mathbb{C}$ or
4 KB (673 words) - 17:01, 26 October 2014
where $A$ is a finite set of integers and $f$ is a real-valued function (cf. also [[Trigonometric sum|Trigonometr
...a finite interval $I \subset \mathbf{R}$. One then takes $A$ as the set of integers $n \in I$ for which $\operatorname{GCD} ( h ( n ) , q ) = 1$, and sets $f (
10 KB (1,528 words) - 07:11, 24 January 2024
A measurable subset $A\subset X$ for which there is an infinite sequence of integers $n_i$ such that the sets $T^{n_i}$ are mutually disjoint (here, invertibili
2 KB (226 words) - 12:04, 23 August 2014
...field of [[rational number]]s is defined by $\nu(a/b) = r$ where $a,b$ are integers and $a/b = p^r \cdot a'/b'$ with $a',b'$ coprime to $p$; set $\nu_p(0) = \i
1 KB (235 words) - 08:06, 26 November 2023
where $n_1,\dotsc,n_k$ are arbitrary integers, while $\alpha_1,\dotsc,\alpha_k$ are given real numbers. This class of fun
2 KB (256 words) - 13:12, 14 February 2020
has a finite number of solutions in rational integers $p$ and $q$, $q>0$, for any $\epsilon>0$, and an infinite number of solutio
...e $m$ is an integer, cannot have more than a finite number of solutions in integers $x$ and $y$.
4 KB (634 words) - 15:17, 14 February 2020
The problem of solving Pythagoras' equation $a^2+b^2=c^2$ in integers $a,b,c$ leads to the [[Pythagorean numbers]]. The problem of solving its ge
2 KB (259 words) - 19:38, 17 December 2016
...oup. Basically, however, one considers only the semi-group of non-negative integers (in other words, one considers iteration of some continuous mapping $T\colo
4 KB (690 words) - 15:52, 14 February 2020
if there exist positive integers $ m $
2 KB (231 words) - 18:48, 5 April 2020
...variable $x$ is piecewise constant (a [[step function]]) with jumps at the integers. Using the integral part one defines the [[fractional part of a number]] $x
2 KB (239 words) - 16:50, 23 November 2023
where $\Phi(x,y)$ denotes the number of positive integers not exceeding $x$ that are free of prime factors smaller than $y$; see [[#R
2 KB (243 words) - 15:57, 22 September 2017
...antity $e^{\psi(x)}$ is equal to the least common multiple of all positive integers $n \le x$. The functions $\theta(x)$ and $\psi(x)$ are related by the ident
2 KB (266 words) - 18:19, 18 October 2014
of non-negative integers with $ i _ {1} + \dots + i _ {k} = n $.
is given (the Milnor–Hirzebruch problem). In other words, which sets of integers $ a _ \omega = a _ {i _ {1} \dots i _ {k} } $,
8 KB (1,150 words) - 18:42, 13 January 2024
i.e. a set of non-negative integers $ i _ {1} \dots i _ {k} $
be a set of non-negative integers, let $ S _ \omega ( p) $
5 KB (680 words) - 08:07, 6 June 2020
In number theory the question of whether integers can be represented as values of forms with integer coefficients for integer
1 KB (237 words) - 07:31, 19 January 2016
...\pmod 3$ if and only if $p$ can be written in the form $p=x^2+27y^2$ with integers $x$ and $y$ (a result conjectured by Euler and first proved by Gauss).
1 KB (215 words) - 20:43, 5 December 2023
...rphic) to $S$, while $\otimes$ denotes the tensor product over the ring of integers, and $(r\otimes s)b = rbs$. For every left $R$-module $M$ one has the situa
2 KB (277 words) - 17:01, 23 November 2023
...llows. Let $\Gamma$ be an infinite class and let $\mu$ and $r$ be positive integers; let all the subclasses of $\Gamma$ which have $r$ elements, in other words
...non-intersecting) subfamilies $T_1,\dots,T_t$ and let $q_1,\dots,q_t,r$ be integers, $q_i\geq r\geq1$, $i=1,\dots,t$. Then there exists a minimal number $n(q_1
4 KB (709 words) - 17:33, 1 November 2014
For any real number $\alpha$ and any natural number $Q$ there exist integers $a$ and $q$ which satisfy the condition
...numbers $\alpha_1,\ldots,\alpha_n$ and any natural number $Q$ there exist integers $a_1,\ldots,a_n$ and $q$ such that
7 KB (1,065 words) - 14:05, 17 March 2020
...up of finite index in $\def\O{\mathcal{O}}G_\O$, where $\O$ is the ring of integers in an algebraic number field $k$ (cf.
If $\def\SL{\textrm{SL}}\G=\SL(n,\Z)$, where $\Z$ is the ring of integers, it was known already in the 19th century that the congruence subgroup prob
4 KB (613 words) - 21:48, 5 March 2012
adic integers, best viewed as the set of formal expansions in powers of $ p $
th root of unity) and its ring of integers $ \mathbf Z _ {p} [ \zeta _ {n} ] $.
7 KB (1,070 words) - 19:40, 5 June 2020
...n integers — Diophantine inequalities — and also with solving equations in integers (cf. [[Diophantine equations|Diophantine equations]]).
...pediaofmath.org/legacyimages/d/d032/d032600/d03260017.png" /> are variable integers (linear homogeneous Diophantine approximations), i.e. the problem of ration
54 KB (7,359 words) - 18:32, 31 March 2017
...pecies]], consisting of finite sequences (cf. [[Tuple|Tuple]]) of positive integers, called the nodes of the spread (or the admissible sequences of the spread)
If one orders the sequences of positive integers by considering that $ \tau < \pi $
5 KB (754 words) - 08:22, 6 June 2020
...wo-person zero-sum game in which the sets of player strategies are sets of integers, while the pay-off function has the form
2 KB (277 words) - 17:56, 25 November 2012
...to J. Wilson. It was proved by J.L. Lagrange in 1771. A primality test for integers follows from Wilson's theorem: A natural number $n>1$ will be prime if and
2 KB (284 words) - 16:21, 18 October 2017
be the set of integers larger than zero with the natural order relation, and let $ G _ {i} = \ma
adic integers.
5 KB (778 words) - 19:48, 21 January 2021
...n of a positive integer $n$ is a decomposition of $n$ as a sum of positive integers. For example, the partitions of 4 read: $4, 3+1, 2+2, 2+1+1, 1+1+1+1$. The
2 KB (231 words) - 15:33, 11 November 2023
...onometric sums, in which the summation variable runs through a sequence of integers, prime numbers, etc. In this way many classical problems in [[Analytic numb
...riables $u$ and $v$ run through the values of (not necessarily successive) integers in respective sets $U$ and $V$, $A<u<2A$. Let $\psi_1(u)$ and $\psi_2(v)$ b
9 KB (1,606 words) - 09:42, 9 November 2014
...ng of algebraic integers|Local-global principles for the ring of algebraic integers]]). That is, it provides a method by which local information can be lifted
8 KB (1,239 words) - 17:01, 1 July 2020
...ite field extension of $\mathbf Q$) is called ''Euclidean'' if its ring of integers $A$ is a [[Euclidean ring]], and ''norm-Euclidean'' if it Euclidean with re
2 KB (243 words) - 18:11, 14 October 2023
...nch of [[Number theory|number theory]] that investigates properties of the integers by elementary methods. These methods include the use of divisibility proper
...egers and on this basis he justifies the main theorem of the arithmetic of integers: Every natural number can be factored in one and only one way into a produc
13 KB (2,043 words) - 20:28, 13 October 2014
for all integers $ n $
for all integers $ n $
5 KB (763 words) - 08:09, 6 June 2020
...ring is the ring of residues modulo $n$ — the quotient ring of the ring of integers $\mathbf Z$ by the ideal $\mathbf Zn$. The elements of $\mathbf Z/\mathbf Z
2 KB (270 words) - 14:56, 30 December 2018
Let $f(q)$ be a non-negative function defined on the positive integers. Gallagher's ergodic theorem, or Gallagher's zero-one law states that the s
2 KB (270 words) - 18:14, 11 April 2016
...$ be the [[free associative algebra]] on $Z = \{Z_1,Z_2,\ldots\}$ over the integers. Give $\mathbf{Z}\langle Z \rangle$ a [[Hopf algebra]] structure by means o
...iel Hazewinkel, "The Algebra of Quasi-Symmetric Functions is Free over the Integers", ''Advances in Mathematics'' '''164''' (2001) 283–300 {{DOI|10.1006/aima
4 KB (709 words) - 13:46, 20 March 2023
are non-negative integers. The weight subspace $ V _ \mu ( \lambda ) $
are non-negative integers.
6 KB (805 words) - 08:11, 6 June 2020
...generalises the construction of the [[rational number]]s from the ring of integers.
2 KB (267 words) - 17:01, 23 November 2023
the integers modulo 2. Let $ x \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _
of non-negative integers such that $ i _ {1} + \dots + i _ {k} = n $.
5 KB (796 words) - 08:23, 6 June 2020
A function defined for all integers $a$ coprime to a given odd integer $P>1$ as follows: Let $P=p_1\dotsm p_r$
2 KB (315 words) - 17:41, 11 November 2023
...tive with respect to addition in the set of real numbers and in the set of integers. A [[distributive lattice]] is a [[lattice]] in which one of the operatio
2 KB (301 words) - 19:27, 24 January 2016
...ct to localizations and extensions of finite type; it includes the ring of integers and all complete local rings. See also [[Excellent ring]].
2 KB (317 words) - 17:54, 10 January 2016
Here, the ring of finite elements is the same as the ring of (integers of) the valuation $ v $.
2 KB (326 words) - 08:06, 6 June 2020
...[[Algebraic number]]) and let $A$ be a set of prime ideals (of the ring of integers $A_K$) of $K$. If an equality of the form
2 KB (310 words) - 09:04, 26 November 2023
-
2 KB (295 words) - 17:43, 19 December 2014
...r, if $M$ is a free finite-dimensional module over the ring $\mathbf Z$ of integers, one speaks of crystallographic groups (cf. [[Crystallographic group|Crysta
2 KB (330 words) - 09:46, 11 August 2014
...nction]] of the numbers $a$ and $p$, defined for odd prime numbers $p$ and integers $a$ not divisible by $p$. The Legendre symbol is denoted by $\left({\frac{a
2 KB (339 words) - 18:03, 19 December 2014
where $\mathfrak{m}$ is the [[maximal ideal]] in the ring of integers of $K$. The smallest number $f$ with this property is called the ''ramifica
2 KB (320 words) - 20:07, 7 September 2017
...problem of the sieve method consists in evaluating for a finite set $A$ of integers the quantity of those elements that are not divisible by any prime number $
4 KB (636 words) - 19:42, 17 November 2023
Thus, $E_{2n+1}=0$, the $E_{4n}$ are positive and the $E_{4n+2}$ are negative integers for all $n=0,1,\dots$; $E_2=-1$, $E_4=5$, $E_6=-61$, $E_8=1385$, and $E_{10
2 KB (297 words) - 11:53, 23 November 2023
The sum of the positive integers divisors of a natural number $n$, including $1$ and $n$:
2 KB (268 words) - 19:41, 17 November 2023
A system consisting of $m$ integers each one of which is the residue of one and only one of the numbers $0,\ldo
2 KB (318 words) - 13:55, 11 August 2014
...omorphic to the $p$-adic Lie group $\mathbb{Z}_{p}$, the group of $p$-adic integers.
2 KB (327 words) - 11:17, 9 April 2023
...hfrak a_i}$, $i=1,\dots,n$. In the particular case when $A$ is the ring of integers $\mathbf Z$, the Chinese remainder theorem states that for any set of pairw
2 KB (355 words) - 13:31, 11 July 2018
The respective integers $\nu ( \lambda )$ and $\operatorname { dim } ( E ( \lambda ) X )$ are calle
2 KB (301 words) - 17:00, 1 July 2020
...tersection with any other non-zero submodule is non-zero. For example, the integers form an essential submodule of the group of rational numbers. Each module i
2 KB (361 words) - 19:20, 2 October 2016
...mathbf Q_p$ be the field of $p$-adic numbers and $\mathbf Z_p$ its ring of integers. Then the quasi-cyclic group for the prime $p$ is the quotient group $\math
2 KB (335 words) - 17:07, 30 July 2014
...me { diag } ( z ^ { k _ { i } } )$ is a diagonal matrix-function with some integers $k_i$ as exponents.
...atorname { diag } ( z ^ { k _ { 1 } } , \ldots , z ^ { k _ { n } } )$. The integers $k_i$, called the partial indices of $f$, are uniquely determined up to the
7 KB (989 words) - 16:56, 1 July 2020
there are integers $ x _ {1}, \ldots, x _ {n} $
2 KB (294 words) - 20:27, 11 November 2023
is the number of integers $ k $
in case of characteristic zero, they are integers. Thus,
4 KB (652 words) - 05:18, 7 March 2022
...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
...construction is possible for angles $m\cdot90^\circ/2^n$, where $n,m$ are integers.
2 KB (321 words) - 13:05, 7 December 2014
Let $f ( q )$ be a function defined on the positive integers and let $\varphi ( q )$ be the Euler [[Totient function|totient function]].
...nal analogue of the Duffin–Schaeffer conjecture: There are infinitely many integers $( p _ { 1 } , \dots , p _ { k } )$ and $q$ such that
10 KB (1,454 words) - 07:42, 4 February 2024
For coprime integers $q$ and $a$, let $\pi(x;a,q)$ denote the number of primes not exceeding $x$
3 KB (458 words) - 16:53, 23 November 2023
...etic function $\phi$ whose value at $n$ is equal to the number of positive integers not exceeding $n$ and relatively prime to $n$ (the "totatives" of $n$). The
2 KB (318 words) - 09:27, 10 November 2023
The problem of finding such a universal method for detecting solutions in integers was posed by D. Hilbert [[#References|[1]]].
...the recognition of the solvability of homogeneous Diophantine equations in integers. This important question is still (1988) open and has not been studied to a
6 KB (845 words) - 19:39, 31 December 2014
...ger solutions. This result (actually in a more general form, for algebraic integers) was established in 1929 by C.L. Siegel [[#References|[1]]], who used the t
...[2]</TD> <TD valign="top"> A.O. Gel'fond, "The solution of equations in integers" , Noordhoff (1960)</TD></TR>
5 KB (784 words) - 20:40, 18 October 2014
...athbf{N} \times \mathbf{N}$, where $\mathbf{N}$ is the set of non-negative integers, with respect to the operation
2 KB (332 words) - 18:22, 16 January 2018
...d $\mathbf Z_p$ are the field of rational numbers and the ring of $p$-adic integers, respectively. The class of the element $a$ is defined as its [[conjugacy c
2 KB (333 words) - 20:58, 29 November 2014
...algebraic equations the solutions of which are sought for in the algebraic integers of some algebraic extension of the field $ \mathbf Q $
Finding solutions of equations in integers is one of the oldest mathematical problems. As early as the beginning of th
18 KB (2,685 words) - 08:57, 18 August 2022
...tion from the arithmetic of the rational integers to the arithmetic of the integers of an extension $ K $
for a pair of integers $ a $
10 KB (1,442 words) - 08:10, 6 June 2020
...ideals play the same role as the prime numbers do in the decomposition of integers in powers of prime numbers, while the primary ideals play the role of power
2 KB (393 words) - 08:07, 6 June 2020
belong to certain sequences of positive integers, $ D ^ { \prime } $
3 KB (352 words) - 17:31, 5 June 2020
...and additional axioms that describe certain data structures, including the integers, ordered pairs, lists, and symbols. The logic also contains a principle of
...that it is built to work on conjectures about recursive functions over the integers and other finitely generated structures only; it does not support, especial
5 KB (832 words) - 18:51, 24 March 2012
into the set of non-negative integers, where $ \nu ( i) = 0 $
2 KB (376 words) - 08:01, 6 June 2020
Multiplication of positive integers is defined in the following way by means of addition: the product of a numb
2 KB (344 words) - 18:24, 30 December 2018
...ign="top">[3]</TD> <TD valign="top"> A.V. Malyshev, "The representation of integers by positive quadratic forms" ''Trudy Mat. Inst. Steklov.'' , '''65''' (1962
3 KB (340 words) - 17:57, 19 October 2014
...of integers of the field $\Q\bigl(\sqrt d\bigr)$ over the ring of rational integers $\Z$, one can take
5 KB (867 words) - 17:41, 12 November 2023
For integers $p=n$ we have
2 KB (361 words) - 11:28, 14 June 2019
Examples. 1) If $R = \mathbf{Z}$ is the ring of integers, then the irreducible $R$-modules are the Abelian groups of prime order. 2)
2 KB (391 words) - 18:11, 18 March 2018
...ero [[Jacobson radical]]. For instance, any [[Artinian ring]], any ring of integers (in general, any [[Dedekind ring]] which is not semi-local) or any [[absolu
3 KB (415 words) - 20:32, 19 January 2016
and for non-negative integers $ r $
2 KB (357 words) - 06:29, 30 May 2020
(where $\mathbf Z_{p} $is the ring of integers of the $p$-adic completion $ \mathbf Q _ {p} $
2 KB (346 words) - 18:45, 11 April 2023
values of the unknowns are numbers (complex, real or integers), but
==Solution of systems of linear equations in integers.==
12 KB (2,085 words) - 22:05, 5 March 2012
A [[recursive sequence]] of integers in which each term is the sum of the two preceding.
Let $P,Q$ be non-zero integers with $\mathrm{hcf}(P,Q) = 1$ and $D = P^2 - 4Q \neq 0$. A ''[[Lucas sequenc
6 KB (943 words) - 20:43, 23 November 2023
There exist positive integers $e_k$ such that the divisor $- \sum _ { k = 1 } ^ { s } e _ { k } D _ { k }
...fibres of $f$. Express $\lambda / r = p / q$ for relatively prime positive integers $p$ and $q$. Then $q \leq r ( d + 1 )$.
6 KB (873 words) - 09:30, 3 February 2024
non-negative parts, i.e., over all solutions in non-negative integers $ c _ {r} $
where the sum is over all solutions in non-negative integers of the equations
12 KB (1,714 words) - 10:58, 29 May 2020
...lgebra|Group algebra]]; [[Cross product|Cross product]]) over the rational integers $ \mathbf Z $.
be the ring of integers of $ K _ {m} $
7 KB (1,035 words) - 05:58, 19 March 2022
...ebraic number]] $\xi$ of degree $n$ there exists a $c>0$ such that for any integers $p$ and $q$ ($q>0$) one has
2 KB (331 words) - 10:10, 13 April 2014
on the set of integers that satisfies the following conditions:
i.e. the system of least non-negative integers satisfying the congruences
10 KB (1,462 words) - 11:49, 26 March 2023
For integers $\nu =n$ the Anger function coincides with the Bessel function $J_\nu$ of o
2 KB (361 words) - 09:02, 22 February 2014
are integers; $ | b _ {n} | = 1 $;
3 KB (392 words) - 17:33, 5 June 2020
for all integers $a$, $b$, $c$, $d$ such that $a d - b c = 1$ and $z$ in the upper half-plan
...$p ( n )$, which is the number of ways of writing $n$ as a sum of positive integers. The discriminant function $\Delta ( z ) = ( 60 G _ { 4 } ) ^ { 3 } - 27 (
13 KB (1,907 words) - 07:36, 22 March 2023
is the ring of integers with a trivial $ G $-
3 KB (405 words) - 18:45, 13 January 2024
...t series over a ring $R$ is associated to a function $a$ from the positive integers to $R$
2 KB (358 words) - 17:25, 11 November 2023
...n {{Cite|Wa}}). If the numbers $\rho+\nu\ge 0$ and $\rho-\nu\ge 0$ are not integers, then
2 KB (333 words) - 06:59, 22 February 2014
is the additive group of integers $ \mathbf Z $),
3 KB (378 words) - 19:39, 5 June 2020
are non-positive integers when $ i \neq j $ ,
...if the corresponding highest weights are the same. Any set of non-negative integers is the set of numerical marks of the highest weight of some finite-dimensio
9 KB (1,307 words) - 20:04, 27 February 2021
...consider the [[free associative algebra]] $\mathbb{Z}[X]$ on $X$ over the integers, provided with the [[Hopf algebra]] structure given by $\mu(X_i) = X_i \oti
2 KB (377 words) - 13:44, 20 March 2023
whose coefficients $a$, $b$ and $c$ are pairwise coprime integers, square-free and not all of the same sign, has a non-zero rational solution
2 KB (327 words) - 14:23, 15 April 2023
Abstractly the problem can be formulated as follows. Given positive integers $c$, $a_1 , \dots , a _ { n }$, $c_ 1 , \ldots , c _ { n }$, the problem is
2 KB (354 words) - 15:30, 1 July 2020
...ts of an algebraic number field $k$ are the units of the ring of algebraic integers of $k$, the $p$-adic units are the units of the ring of [[P-adic number|$p$
2 KB (403 words) - 19:25, 3 April 2016
...f. also [[Projective module|Projective module]]). Examples are the ring of integers, the ring of polynomials in one variable over a field, von Neumann regular
3 KB (403 words) - 17:47, 27 August 2014
...tangles, then $r ^ { i } ( A ) * r ^ { j } ( B )$ is $n$-algebraic for any integers $i$, $j$, where $r$ denotes the rotation of a tangle by the angle $\pi /n$
2 KB (401 words) - 16:58, 1 July 2020
...l identities $p[x_1,\ldots,x_m]=0$ of the $n\times n$ matrix ring over the integers. The condition is sufficient if $R$ is prime or semi-prime, but fails in ot
5 KB (851 words) - 14:57, 30 July 2014
...hfrak{p}_K^n$, where $\mathfrak{p}_K$ is the maximal ideal of (the ring of integers $A_K$ of) $K$ and $n$ is the smallest integer such that $N_{L/K} L^* \subse
2 KB (440 words) - 19:42, 7 March 2018
...\alpha$ is an algebraic number of degree $n\geq3$, $x>0$, $y$ are rational integers, and $c>0$ and $\delta>0$ are explicitly defined in terms of $\alpha$ [[#Re
...butions to the theory of Diophantine equations I: on the representation of integers by binary forms" ''Phil. Trans. Royal Soc. London Ser. A'' , '''263''' (1
6 KB (940 words) - 18:12, 23 November 2014
where $n _ { 1 } , \ldots , n _ { k }$ is a decreasing sequence of positive integers. This representation cannot include two consecutive Fibonacci numbers, say
...a2]]], where there is a discussion of algorithms which, given two positive integers $a$ and $b$ in Zeckendorf form, produce the Zeckendorf forms for $a + b$ an
7 KB (975 words) - 17:43, 1 July 2020
...s follows. Let $\lambda(n)$ be the exponent of the multiplicative group of integers modulo $n$, that is, the least positive $\lambda$ making all $\lambda$-th p
3 KB (485 words) - 17:34, 18 October 2014
...ed that the $U _ { n } ( x )$ are irreducible polynomials over the ring of integers if and only if $n$ is a prime number (cf. also [[Irreducible polynomial|Irr
where the sum is taken over all non-negative integers $n _ { 1 } , \ldots , n _ { k }$ such that $n _ { 1 } + 2 n _ { 2 } + \ldot
14 KB (2,016 words) - 07:46, 27 January 2024
The non-negative integers with the usual operations provide an example of a semi-ring that does not s
2 KB (371 words) - 05:54, 15 April 2023
...grally closed (cf. also [[Integral domain|Integral domain]]). The rings of integers of number fields are completely integrally closed, and hence, Fatou rings.
2 KB (407 words) - 19:17, 19 April 2012
...definition, the extension has a normal integral basis provided the ring of integers $O _ { \text{N} }$ is a free $\mathbf{Z}[ \text{Gal} (N/K)]$-module).
...tegral basis. There are two different notions: One can require the ring of integers $O _ { \text{N} }$ to be a free $O _ { K } [G]$ module (necessarily of rank
12 KB (1,811 words) - 17:46, 1 July 2020
in countably many commuting variables over the integers and its subring
defined by $w$. These form a basis over the integers of $\mathbf{Q}^{\mathrm{sym}}_{\mathbf{Z}}(X))$.
4 KB (708 words) - 13:51, 20 March 2023
...l system|dynamical system]] defined by the action of the additive group of integers $ \mathbf Z $ (or the additive semi-group of natural numbers $ \mathbf
3 KB (418 words) - 05:47, 18 May 2022
are sets of integers, then the function $ f $
3 KB (437 words) - 08:11, 6 June 2020
the ring of integers, the definition coincides with the usual definition of the rank of an Abeli
3 KB (439 words) - 08:09, 6 June 2020
...I$ or $a^n \in I$ for some natural number $n$. In the ring $\mathbf{Z}$ of integers a primary ideal is an ideal of the form $p^n\mathbf{Z}$, where $p$ is a pri
3 KB (455 words) - 19:51, 5 October 2017
...of logarithm tables, which contain only the mantissa of the logarithms of integers.
3 KB (453 words) - 15:11, 19 August 2014
...fact that other input and output formats can be encoded into non-negative integers.
...ithm for computing a function from non-negative integers into non-negative integers. Denote by $MC_i(j)$ the value of the function computed by $MC_i$ for the a
13 KB (2,014 words) - 20:35, 31 December 2018
...malization of the vectors $X_\alpha$ under which all $N_{\alpha\beta}$ are integers (see [[Chevalley group|Chevalley group]]). The definition of a Cartan–Wey
3 KB (465 words) - 10:03, 23 August 2014
for any pair of [[coprime numbers|coprime integers]] $m,n$. It is usually assumed that $f$ is not identically zero (which is
3 KB (419 words) - 20:15, 19 November 2017
...that every natural number is the sum of $s$ $k$-th powers of non-negative integers. D. Hilbert in 1909 was the first to give a general solution of Waring's pr
in non-negative integers. Hilbert's theorem then states that there exists a $s=s(k)$ for which $J_{s
6 KB (925 words) - 14:27, 12 November 2023
A relation between two integers $ a $
Therefore, the relation " (modm)" divides the set of all integers into non-intersecting equivalence classes $ A,\ B ,\dots $.
20 KB (3,011 words) - 09:59, 26 March 2023
...ct group of a block]]). Let $\nu$ be the discrete valuation defined on the integers with $\def\a{\alpha}\nu(np^\a)=\a$ whenever $n$ is prime to $p$. By a theor
2 KB (352 words) - 14:21, 13 April 2012
...{ p } = A _ { K } \cap \mathfrak { P }$, where $A _ { K }$ is the ring of integers of $K$. There is a unique element
3 KB (449 words) - 17:00, 1 July 2020
Such a diagram can be filled with integers from $1$ to $n$ in increasing order in each row and in each column. This is
3 KB (395 words) - 19:47, 9 November 2023
...s. All [[cyclic group]]s are Abelian, in particular, the additive group of integers. All [[direct sum]]s of cyclic groups are Abelian. Also the additive group
...\dots,g_k$ in an Abelian group is called linearly dependent if there exist integers $n_1,\dots,n_k$, not all equal to zero, such that $\sum_{i=1}^kn_ig_i=0$. I
11 KB (1,810 words) - 22:12, 29 August 2015
...number field containing the $n$-th roots of unity. Let $A$ be the ring of integers of $K$ and let $\mathfrak{p}$ be a prime ideal of $A$. Let $\mathfrak{p}$ b
3 KB (435 words) - 19:32, 19 December 2014
are rational integers. The modular group can be identified with the quotient group $ \mathop{\
real numbers (respectively, integers) and $ ad - bc = 1 $.
7 KB (1,031 words) - 18:33, 13 January 2024
where the $\alpha_k$ are integers between 0 and $q-1$ (inclusive). In the representation of $x$ by a $q$-ary
3 KB (412 words) - 14:41, 7 March 2019
...on an interval $I$, is completely monotonic on $I$ if for all non-negative integers $n$,
4 KB (582 words) - 12:38, 23 February 2024
...$C=\partial D\subset\partial K$. For each pair $(k_+,k_-)$ of non-negative integers, not both $0$, there is a unique oriented diffeomorphism type of kinky hand
...the same [[#References|[a1]]], [[#References|[a7]]]. For any non-negative integers $k_\pm$, not both $0$, there is a Casson handle with $T_1$ having exactly $
7 KB (1,047 words) - 15:29, 4 October 2014
where the parameters <math>N,M,m</math> are non-negative integers which satisfy the condition <math>m\leq M\leq N</math>. A negative hypergeo
3 KB (447 words) - 14:48, 5 June 2017
are integers, $ 0 < \alpha \leq 1 $,
3 KB (472 words) - 22:11, 5 June 2020
...ts,p_k$ distinct prime numbers and $\alpha_1,\ldots,\alpha_k$ non-negative integers, is isomorphic to the direct product of distributive quasi-groups $Q_1,\ldo
3 KB (425 words) - 22:16, 7 January 2017
runs through the real numbers or the integers). It consists of the existence of a trajectory $ \{ T _ {t} w _ {0} \} $
3 KB (470 words) - 08:25, 6 June 2020
...$ using multiplication, for any system $U,U_1,\ldots,U_q$ of matrices over integers. The case in which $U,U_1,\ldots,U_q$ are square matrices of the same order
3 KB (462 words) - 20:23, 31 July 2014
...containing all infinite ones, let $\mathcal{O} _ { S }$ be the ring of $S$-integers and let $\mathcal{O} _ { S } ^ { * }$ be the group of $S$-units of $K$. Let
...valign="top"> N.P. Smart, "Thue and Thue–Mahler equations over rings of integers" ''J. London Math. Soc.'' , '''56''' : 2 (1997) pp. 455–462</td></tr>
7 KB (1,096 words) - 17:46, 1 July 2020
...For instance, the ordinal number of the set $ \mathbf{N} $ of all positive integers, ordered by the relation $ \leq $, is $ \omega $. The ordinal number of the
...rdinal numbers in a form resembling the decimal representation of positive integers. If $ \gamma > 1 $ and $ 1 \leq \alpha < \gamma^{\eta} $, then there exists
9 KB (1,404 words) - 18:33, 4 December 2017
are called algebraic integers. If a ring $ B $
3 KB (562 words) - 22:12, 5 June 2020
...essence may be explained as follows. Consider a given sequence of positive integers $n_1,\ldots,n_Z$ not larger than $N$, a prime number $p < \sqrt N$ and a re
3 KB (418 words) - 21:29, 11 November 2017
the greatest common divisor of all the integers $ k $
3 KB (442 words) - 10:33, 29 May 2020
...A [[lattice]] $L$ is isomorphic to a subset of the totally ordered set of integers if and only if every sublattice of it is a [[Retract|retract]].
3 KB (509 words) - 06:25, 28 March 2023
...also [[Algebraic number|Algebraic number]]; [[Field|Field]]) with ring of integers $ R $
4 KB (600 words) - 17:32, 5 June 2020
...s the number of executed operations. (In particular, for computations over integers the Turing model uses a logarithmic size measure together with bit-costs wh
...near programming|linear programming]] problem is in $\mathcal{P}$ over the integers in the Turing model (see [[#References|[a4]]]). It is not yet (1998) known
7 KB (1,161 words) - 16:46, 1 July 2020
be the ring of integers of $ K $,
4 KB (569 words) - 05:59, 11 October 2023
at these points be non-integers. Let $ a $
3 KB (479 words) - 17:43, 16 December 2020
are algebraic integers. Every character of a group $ G $
for some integers $ a _ {1} \dots a _ {n} $.
10 KB (1,488 words) - 19:39, 5 June 2020
...$-group]]s. E.g. the additive group of the ring $\mathbf{Z}_p$ of $p$-adic integers is a pro-$p$-group. In Galois theory pro-$p$-groups appear as [[Galois grou
3 KB (535 words) - 19:45, 21 October 2017
...ch of number theory with the basic aim of studying properties of algebraic integers in algebraic number fields $ K $ of finite degree over the field $ \ma
...x _{n} \omega _{n} $ where all the $ x _{i} $ run through the rational integers (i.e. $ \mathbf Z $ ). Moreover, such a representation is unique for each
28 KB (4,440 words) - 22:00, 11 December 2019
A method for adding sequences of positive integers; created by L.G. Shnirel'man in 1930. Let $ \nu ( x) \neq 0 $
3 KB (458 words) - 19:05, 20 November 2023
...two complex constants, $\operatorname{Im}c=0$; $m\geq2$, and $n\geq1$ are integers named, respectively, the order and the index of the singular point and $u$
3 KB (495 words) - 13:00, 3 October 2014
...als]] evaluated at rational arguments: Let $h$ and $k$ be relatively prime integers. If $n$ is even, then
...{ 1 } + 1 / 2$. It has also been shown [[#References|[a10]]] that for all integers $h$, $k$, $n$ with $k \neq 0$ and $n \geq 1$, $k ^ { n } ( B _ { n } ( h /
8 KB (1,206 words) - 16:55, 1 July 2020
the ring of integers, and also quotient rings of them, are Noetherian. Every [[Artinian ring]] i
3 KB (539 words) - 18:51, 3 April 2024
its ring of integers, $ B $
the maximal order is the ring of integers of $ B $,
13 KB (2,001 words) - 02:12, 1 March 2022
...ymmetric way. The classical semi-simple rings and all rings of residues of integers $ \mathbf Z /( n) $
4 KB (550 words) - 08:13, 6 June 2020
of non-negative integers satisfying the condition $ \lambda _ {1} \geq \dots \geq \lambda _ {n}
4 KB (519 words) - 16:18, 5 February 2022
that is defined for any set of non-negative integers $ n _ {1} \dots n _ {k} $
4 KB (522 words) - 13:11, 6 January 2024
are not integers and the relevant derivatives are continuous, then a Nikol'skii space is a [
if and only if for any non-negative integers $ k $
11 KB (1,572 words) - 19:46, 12 January 2024
...If $p(x)=x^2$ or $p(x)=x^3$, one obtains sequences of squares or cubes of integers, i.e. special cases of arithmetic series of the second and third orders. If
4 KB (566 words) - 07:28, 26 March 2023
3) if $k$ and $n$ are integers which satisfy $2^n\leq k<2^{n+1}$, then $w_k$ changes sign once on interval
3 KB (467 words) - 09:42, 27 November 2018
functors of their rings of integers, and also between the values of the $ L $-
...<TD valign="top"> B. Harris, G. Segal, "$K_i$ groups of rings of algebraic integers" ''Ann. of Math.'' , '''101''' : 1 (1975) pp. 20–33 {{MR|0387379}} {{ZBL|
8 KB (1,120 words) - 20:06, 31 October 2023
is a collection of non-negative integers, then $ c _ \omega $
be a set of non-negative integers. Let $ S _ \omega ( c _ {1} \dots c _ {n} ) $
13 KB (1,910 words) - 19:40, 7 January 2024
...D valign="top"> J.H. Conway, N.J.A. Sloane, "Self-dual codes over the integers modulo $4$" ''J. Combin. Th. A'' , '''62''' (1993) pp. 31–45</TD></TR>
3 KB (459 words) - 17:32, 17 April 2016
runs over all non-negative integers less than $ k $
3 KB (502 words) - 14:27, 17 March 2023
...and $k,l,\dots,t$ (the ''exponents of the powers'', which are non-negative integers) are constants. The individual expressions
...of polynomials similar to that played by the prime numbers in the ring of integers. For example, the following theorem holds: If a product $PQ$ is divisible b
9 KB (1,497 words) - 10:44, 27 June 2015
are integers, and $ n $
3 KB (485 words) - 19:43, 5 June 2020
with coefficients in the ring of integers $ \mathbf Z $
5 KB (636 words) - 22:11, 5 June 2020
Positive integers are written by repeating these numerals. When a letter is followed by a let
11 KB (1,241 words) - 08:12, 6 June 2020
...$A$ and $B$ of finite $k$-dimension are equivalent if there exist positive integers $m$ and $n$ such that the tensor products $A \otimes_k M_m(k)$ and $B \otim
...ditive group of rational numbers and $\mathbb{Z}$ is the additive group of integers. This fact is of importance in local class-field theory.
7 KB (1,232 words) - 12:12, 30 December 2015
the ring of integers, then mod- $ R $
4 KB (572 words) - 08:01, 6 June 2020
are non-negative integers and $ M \leq N $,
4 KB (602 words) - 22:11, 5 June 2020
...[2]</TD> <TD valign="top"> A. Mostowski, "On definable sets of positive integers" ''Fund. Math.'' , '''34''' (1947) pp. 81–112</TD></TR></table>
4 KB (548 words) - 15:41, 22 June 2020
...rithmetic theory of cubic forms over [[number field]]s (and their rings of integers) is still (1987) rather poorly developed in comparison with the rich and me
3 KB (526 words) - 11:44, 12 October 2023
runs through the set of integers and each $ b _ {k} \in A $.
4 KB (566 words) - 10:58, 29 May 2020
...very ring can naturally be considered as an operator ring over the ring of integers.
4 KB (634 words) - 07:02, 30 March 2024
...indexed by $i$ running over the natural numbers (one-sided, superword) or integers (two-sided, bi-infinite word).
3 KB (544 words) - 07:12, 10 November 2023
are zero or negative integers, the series (2) terminates after a finite number of terms, and the hypergeo
are integers, can be obtained by iterated application of Gauss' relations. The following
12 KB (1,576 words) - 01:14, 21 January 2022
...dditive group of a graded algebra (considered as a module over the ring of integers) is a positively [[Graded module|graded module]]. As an example of a graded
4 KB (576 words) - 19:42, 5 June 2020
...uares: Any natural number can be represented as the sum of four squares of integers. This was established by J.L. Lagrange [[#References|[1]]]. For a generaliz
4 KB (562 words) - 21:05, 11 October 2014
where $g(c_i)$ and $h(c_i)$ are integers. Thus, $g(c_i)$ divides $f(c_i)$. Choosing arbitrary divisors $d_i$ of the
3 KB (574 words) - 18:14, 14 June 2023
...m p ^ { r - 1 } } ( \operatorname { mod } p ^ { 3 r } )$ for all positive integers $m$, $r$ and all prime numbers $p \geq 5$. Another congruence is $a _ {( p
3 KB (525 words) - 20:51, 23 January 2024
...sums actually lie in the ring $\mathbf{Z} [ \zeta _ { e } ]$ of algebraic integers in the cyclotomic field ${\bf Q} [ \zeta _ { { e } } ]$ (cf. also [[Cyclot
<tr><td valign="top">[a8]</td> <td valign="top"> B. Schmidt, "Cyclotomic integers and finite geometry" ''J. Amer. Math. Soc.'' (to appear)</td></tr>
9 KB (1,331 words) - 19:36, 13 February 2024
are integers (a special case of Abel's theorem, see [[Abelian function|Abelian function]
are integers such that $ \alpha \delta - \beta \gamma = n $,
9 KB (1,292 words) - 19:08, 20 January 2022
4 KB (577 words) - 10:26, 10 December 2023
...hat of a diagonalizable group. If $M={\mathbb Z}$ is the additive group of integers, then $D_S(M)$ coincides with the multiplicative group scheme $G_{m,S}$.
4 KB (629 words) - 20:08, 15 December 2020
...one of the following sets: the set of natural numbers, the set of rational integers, the set of integral ideals of a given algebraic number field, a lattice in
4 KB (608 words) - 08:18, 4 November 2023
In number theory one considers equations over the integers, the study of the solutions of which forms the subject of the theory of [[D
4 KB (692 words) - 13:38, 31 July 2014
...whose states form a single zero class is the symmetric random walk on the integers:
4 KB (466 words) - 07:59, 6 June 2020
...ng of algebraic integers|Local-global principles for the ring of algebraic integers]]).
15 KB (2,309 words) - 06:58, 13 February 2024
...ic to the field $\mathbb{Z}/p\mathbb{Z}$ of residue classes of the ring of integers modulo $p$. In any fixed [[Algebraic closure|algebraic closure]] $\Omega$ o
4 KB (749 words) - 18:32, 2 November 2014
there exist integers $ q _ {i} $,
4 KB (615 words) - 22:15, 5 June 2020
are non-negative integers,
4 KB (524 words) - 04:11, 6 June 2020
adic integers (for any prime $ p $);
4 KB (674 words) - 17:31, 5 June 2020
of integers.
4 KB (643 words) - 19:38, 5 June 2020
...$( X _ { n } ) _ { n \in {\bf Z} ^ { d }}$, indexed by the $d$-dimensional integers ($d \geq 1$). The definition of $\mathcal{T}$ is the same as before, but no
4 KB (552 words) - 16:56, 1 July 2020
...lpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $z = ( z_ 1 , \dots , z _ { n } )$, $z ^ { \alpha } = z _ { 1 } ^ { \alpha
7 KB (1,065 words) - 09:52, 11 November 2023
...dius $\epsilon > 0$ centred at $a + i b$, where $a$, $b$ range over all integers, satisfies this criterion and hence is in $\operatorname{BMOA}$.
5 KB (707 words) - 19:01, 22 January 2024
...pute functions defined on an infinite domain of constructive objects (e.g. integers, words over a finite alphabet, finite graphs, infinite trees, etc., cf. [[C
3) Abstract computers which usually process words or integers and have an elaborate structure determined by practical requirements, havin
9 KB (1,295 words) - 17:46, 4 June 2020
a discretely valued complete field; the ring of integers is $ k[[ X ]] $,
8 KB (1,203 words) - 10:36, 20 January 2024
is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] of the group of integers $ \mathbf Z $
4 KB (616 words) - 19:33, 20 January 2021
are integers, is solvable, and even gave an asymptotic formula for the number of solutio
5 KB (759 words) - 08:29, 6 June 2020
is the group of integers under addition (cf. also [[Homology|Homology]]). A group which satisfies th
4 KB (665 words) - 08:29, 6 June 2020
are integers with $ n $
4 KB (623 words) - 08:03, 6 June 2020
...ther than numbers; thus, it does calculations in exact mode. For instance, integers are regarded as strings of digits of arbitrary length and $\sqrt 3$ is rega
...s handled by these systems are polynomials over effective rings (e.g., the integers, the rational numbers, algebraic number fields, finite rings), strings of c
9 KB (1,292 words) - 20:51, 18 September 2016
...inite alphabet $A$ and assume a [[total order]] on the letters. For given integers $p$ and $n$, every sufficiently long word in $A^*$ has either a factor whic
3 KB (562 words) - 14:15, 12 November 2023
...prime number theorem]]), and the construction of a theory of quotients of integers that is essentially equivalent to the theory of rational numbers. With the
9 KB (1,351 words) - 20:43, 26 November 2016
...bers from the [[Uniform distribution|uniform distribution]]: A sequence of integers is initialized with a value $z_0$ and continued as
...ncrement, and $z_0$, the starting number, are suitably chosen non-negative integers. Three choices of $m$, $a$ and $r$ are common on most computers:
11 KB (1,696 words) - 19:30, 8 February 2024
...is the quotient group of the group of all real numbers by the subgroup of integers, and is the largest of all homological dimensions.
coincides with the cohomological dimension for the group of integers.
12 KB (1,894 words) - 19:35, 5 June 2020
Examples of principal ideal rings include the ring of integers, the ring of polynomials $ F ( x) $
5 KB (880 words) - 19:00, 9 January 2024
exist for all non-negative integers $ \nu _ {1} \dots \nu _ {k} $
5 KB (661 words) - 12:22, 17 January 2021
are integers. The dimension polynomial is a birational invariant of the field, that is,
5 KB (651 words) - 08:36, 1 July 2022
Suppose that a set of integers $ ( d _ {1} \dots d _ {n} ) $
6 KB (931 words) - 20:20, 15 March 2023
for all but a finite number of integers $ n $.
6 KB (913 words) - 19:44, 16 January 2024
$n>1$, is solvable (or unsolvable) in integers. Questions on the distribution of power residues and non-residues have been
4 KB (628 words) - 19:38, 19 December 2014
...unity if and only if $|z|=1$ and $\arg z=2\pi m/n$, where $m$ and $n$ are integers, i.e. if and only if
4 KB (680 words) - 13:40, 30 December 2018
...ginated in [[#References|[a1]]], Chapt. II, Sect. 3. Denoting the positive
integers by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
10 KB (1,395 words) - 06:44, 9 October 2016
be positive integers. If $ a _{1} \dots a _{s} $
...with zero-sum solutions to systems of equations and inequalities over the integers. Surveys of this can be found in [[#References|[a2]]] and [[#References|[a4
10 KB (1,573 words) - 17:25, 28 January 2020
...problems is as follows [[#References|[4]]]: Uniformly with respect to the integers $ k \geq 2 $,
for any integers $ k \geq 2 $,
11 KB (1,578 words) - 11:58, 26 March 2023
...f matrices $(a_{ij})\in \GL_n(\Z_p)$ (where $\Z_p$ is the ring of $p$-adic integers), such that $a_{ij}\in p\Z_p$ for $i>j$, and let $N$ be the subgroup of mon
4 KB (647 words) - 09:57, 30 January 2014
...last theorem is the claim that $x^n+y^n=z^n$ has no solutions in non-zero integers for $n>2$. For $n=2$ there is an infinity of solutions, such as: $3^2+4^2=5
...$\mathcal E_{a,b,c}$ given by $y^2=x(x-a)(x+b)$, where $a+b=c$ in non-zero integers. This Frey curve $\mathcal E_{a,b,c}$ is an [[Elliptic curve|elliptic curve
8 KB (1,419 words) - 15:20, 17 March 2023
...rs for the module of all homogeneous solutions). Examples are the rings of integers and of rational numbers, algebraic number fields, and finite rings. For suc
10 KB (1,471 words) - 14:52, 30 May 2020
...ps is of course the multiplicative semi-group $\mathbb{N}$ of all positive integers $\{1,2,3,\ldots\}$, with its subset $P_{\mathbb{N}}$ of all rational prime
...K$ of all non-zero ideals in the [[Ring|ring]] $R = R(K)$ of all algebraic integers in a given [[Algebraic number|algebraic number]] field $K$, with $|I| = \ma
24 KB (3,738 words) - 07:41, 7 February 2024
...n a union of subsets $B _ { \kappa }$, labelled by unordered $n$-tuples of integers $\kappa$. Each of these consists of all loops with $\kappa$ as the set of p
4 KB (630 words) - 17:01, 1 July 2020
...one resulting from the binomial theorem) without having to use division by integers.
4 KB (663 words) - 08:28, 20 January 2024
...r symmetric bilinear, form over an ordered field is a pair of non-negative integers $ ( p, q) $,
4 KB (651 words) - 18:47, 13 January 2024
...ulated as follows: Let $J_{k,n}(N)$ be the number of solutions in positive integers $x_1,\ldots,x_n$ of the equation
...r $n=2$ by showing that any positive integer is the sum of four squares of integers. D. Hilbert (1909) was the first to give a general solution to Waring's pro
27 KB (4,516 words) - 18:38, 18 October 2014
be integers. Then a field $ F $
...|non-standard analysis]]; in particular, non-standard models of the reals, integers, etc. can be obtained as ultrapowers of $ \mathbf R $,
10 KB (1,605 words) - 16:57, 13 January 2024
are non-negative integers, and $ a _ {1} \dots a _ {p} $,
5 KB (645 words) - 08:01, 6 June 2020
...t and output of arithmetic, logical and text data. Arithmetic data include integers, real (with ordinary and enhanced precision) and usually also complex numbe
4 KB (631 words) - 19:39, 5 June 2020
respect to the norm of $\fp$, and let $A_\fp$ be the ring of integers of
4 KB (757 words) - 21:21, 22 November 2014
algebraic integers of an imaginary quadratic field $k$ and $f\in {\mathbb N}$; or
is a subring of finite index in the ring of algebraic integers of the
19 KB (3,251 words) - 20:37, 19 September 2017
...nd a family of them. When a NEF is concentrated on the set of non-negative integers, its elements are sometimes called "power series" distributions, since th
...) ^ { p }$ (see [[#References|[a4]]]. Trivially, it contains all positive integers. The model
12 KB (1,855 words) - 17:45, 1 July 2020
are non-negative integers and $ \alpha _ {0} + \dots + \alpha _ {k} = m $,
5 KB (717 words) - 08:28, 6 June 2020
is simply the ordered set of integers $ (k _{1} \dots k _{r} ) $ ,
5 KB (737 words) - 10:19, 17 December 2019
be a sequence of positive integers. From $ \mathbf n $
4 KB (631 words) - 08:14, 6 June 2020
...alpha_i$ are the principal values of the logarithms. If there are rational integers $b_1,\ldots,b_n$, $|b_i| < B$, such that
5 KB (776 words) - 08:31, 23 November 2023
...laim that $x ^ { n } - y ^ { n } = z ^ { n }$ has no solutions in non-zero integers for $n > 2$ (see also [[Fermat last theorem|Fermat last theorem]]). Howe
4 KB (702 words) - 16:45, 1 July 2020
3) If $a$ and $b$ are integers, $b\ge 1$, if $r$ is a real number, and if
...expansion of a number $r$ into a regular continued fraction, then for any integers $a$ and $b$ it follows from $b>0$, $\d_n \ne a/b$ and
11 KB (1,649 words) - 11:41, 4 September 2013
...\dim X=\dim_\mathbf ZX$ if $\dim X<\infty$ and $\mathbf Z$ is the group of integers. It was subsequently noted by L.S. Pontryagin that homological dimensions w
5 KB (741 words) - 06:05, 22 November 2018
...all [[#References|[a24]]] proved that there exist infinitely many abundant integers $n \equiv a ( \operatorname { mod } b )$ (with $a$ and $b$ given). Let $k$
...expressed as the sum of two abundant numbers. Actually, this is valid for integers $> 20162$, see [[#References|[a27]]].
15 KB (2,194 words) - 05:54, 15 February 2024
be the ring of integers of an algebraic number [[Field|field]] $ F $(
5 KB (731 words) - 08:21, 26 March 2023
are integers and $ \Delta _ {j} $
is the constant sheaf of integers. Thus, if $ H ^ {1} ( M, {\mathcal O} ) = H ^ {2} ( M, \mathbf Z ) = 0
16 KB (2,209 words) - 11:03, 26 March 2023
...s/s085/s085000/s0850009.png" /> and if there exists a sequence of rational
integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
16 KB (2,130 words) - 07:52, 11 December 2016
into the ring of integers $ \mathbf Z $.
5 KB (659 words) - 06:06, 23 April 2024
are integers, namely, the multiplicities with which the $ \pi _ \alpha $
6 KB (828 words) - 09:30, 26 March 2023
...{ n } + a _ { 1 } x ^ { n - 1 } + \dots + a _ { n }$, where the $a_k$ are integers, $a _ { n } \neq 0$ and the roots of $P ( x )$ other than $\theta$ all lie
5 KB (723 words) - 19:03, 23 January 2024
and the integers $ n > 0 $,
6 KB (838 words) - 08:08, 6 June 2020
in the sense of Weyl). For integers $ r = 1, 2 \dots $
5 KB (791 words) - 10:58, 29 May 2020
The simplest algebraic operations — arithmetic operations on positive integers and positive rational numbers — can be encountered in the oldest mathemat
...e best textbooks of that time, L. Euler's Introduction to algebra includes integers, ordinary and decimal fractions, roots, logarithms, algebraic equations of
17 KB (2,478 words) - 16:09, 1 April 2020
composed of the integers from 1 up to $ n ^ {2} $
5 KB (762 words) - 19:27, 12 January 2024
...ng machines computing functions from non-negative integers to non-negative integers.) Let $ T $
16 KB (2,413 words) - 16:10, 1 April 2020
...n of a positive integer $n$ is a decomposition of $n$ as a sum of positive integers. For example, the partitions of $4$ read: $4$, $3+1$, $2+2$, $2+1+1$, $1+1+
4 KB (718 words) - 15:32, 11 November 2023
which runs over the non-negative integers, is called the dimension of $ H ^ { n } ( G, A) $.
For arbitrary integers $ m $
16 KB (2,427 words) - 09:48, 26 March 2023
adic integers, the ring $ k [[ T ]] $
5 KB (800 words) - 19:36, 5 June 2020
is the completion of the ring of integers $ \mathbf Z $
with Galois group isomorphic to the group of $ l $-adic integers $ \mathbf Z _ {l} $.
12 KB (1,769 words) - 11:19, 26 March 2023
...sot $E$-sequence $E ( a _ { 0 } , a _ { 1 } )$ is the sequence of positive integers $\{ a _ { n } \}$ defined for $0 < a _ { 0 } < a _ { 1 }$ by the recursion
5 KB (774 words) - 20:10, 24 January 2024
be integers such that
Ramsey's theorem: Let there be given integers $ q _ {1} \dots q _ {l} $
13 KB (1,942 words) - 08:13, 6 June 2020
...power of 2. In this connection, a (single) arithmetic error in the ring of integers $ \mathbf Z $
8 KB (1,189 words) - 09:46, 26 March 2023
...of the binomial coefficients and the Newton binomial formula for positive integers $ n $
...L. Euler. In his papers on the partitioning and decomposition of positive integers into summands he laid down the beginnings of one of the basic methods of ca
28 KB (4,167 words) - 17:45, 4 June 2020
...e equation $x^n+y^n=z^n$ (the Fermat equation) has no solution in non-zero integers $x,y,z$. It was stated by P. Fermat in about 1630 in the margins of his cop
has no solutions in non-zero relatively-prime integers $x,y,z$. One can also assume that $x$ and $y$ are relatively prime to $p$.
16 KB (2,596 words) - 09:27, 13 February 2024
...cal N P$-hard (cf. [[NP|$\cal N P$]]), even if distances are rounded up to integers and it is required only to decide whether a tour exists whose total length
4 KB (704 words) - 12:30, 17 February 2021
are integers $ \geq 1 $,
6 KB (821 words) - 01:34, 26 April 2022
...]] for the relation between the divisor class group of a ring of algebraic integers and Abelian field extensions.
5 KB (820 words) - 19:36, 5 June 2020
be positive integers such that $ 0 \leq n \leq s \leq 2n - 1 $.
8 KB (1,138 words) - 09:31, 19 January 2021
...oup of class two is uniquely determined by its group ring over the ring of integers, and that a countable Abelian $p$-group is uniquely determined by its group
5 KB (872 words) - 22:00, 29 April 2012
are integers, and $ u _ {m} ^ {n} $
6 KB (908 words) - 14:48, 7 April 2020
are mutually prime integers and $ q \geq 2 $,
6 KB (861 words) - 13:02, 6 January 2024
are integers.
5 KB (794 words) - 19:38, 5 June 2020
...-groups) need not have minimal ideals (the simplest example is the ring of integers) and a minimal ideal in a ring, if it exists, need not be unique. The sum o
6 KB (961 words) - 18:14, 7 May 2016
...an arbitrary [[Polynomial|polynomial]] over a [[Field|field]] or over the integers can be factored into irreducible factors, essentially uniquely (cf. also [[
...m extremely well, both in theory and practice. Of course, factorization of integers and thus also in $\mathbf Z [ x ]$ remains difficult; much of cryptography
12 KB (1,739 words) - 13:11, 26 March 2023
...the so-called Kloosterman sums, which appear in the study of partitions of integers and in the more general Hardy–Littlewood–Ramanujan methods. From the po
but also appear in complex-conjugate pairs. Furthermore, as algebraic integers, they are stable under the action of the Galois group of the rational numbe
12 KB (1,769 words) - 08:31, 26 March 2023
on the positive integers. For example, for irrational $ \alpha $
6 KB (911 words) - 12:00, 22 December 2019
...nce to $\mu_k(f)$. There exists an absolute constant $C$ such that for all integers $k$ and all integer-valued additive arithmetic functions $f(m)$ such that $
...ore precise information on $\omega(n)$. Denote by $N(x;a,b)$ the number of integers in the interval $[3,x]$ for which the inequalities
10 KB (1,718 words) - 04:55, 8 August 2018
is the group of integers. Furthermore, if the polyhedron is finite, the integer group, which is an A
6 KB (897 words) - 08:54, 25 April 2022
...as the discriminant of the lattice $M$. If $M$ coincides with the ring of integers of the field $K$, the discriminant of $M$ is simply called the discriminant
...be a finite separable field extension. Let $A_F$ and $A_E$ be the rings of integers (principal orders) of $F$ and $E$, respectively. Let $\def\fm{ {\mathfrak m
16 KB (2,947 words) - 08:53, 9 December 2016
is the ring of integers in the imaginary quadratic number field $ \mathbf Q ( \sqrt {- d } ) $.
5 KB (775 words) - 07:59, 26 March 2023
and not merely positive integers). If one chooses for $ X $
7 KB (1,086 words) - 17:45, 4 June 2020
are integers.
8 KB (976 words) - 17:46, 4 June 2020
Let $S$ be a fixed set of integers. Let $b _ { j }$ be fixed complex numbers and let $z_j$ be complex numbers
8) Given a finite set $S$ of integers, fixed complex numbers $b _ { j }$, $d \in [ 0,3 ]$, and two generalized po
40 KB (5,729 words) - 17:51, 5 May 2024
...th the $p$-adic valuation $\nu$ with $\nu(p)=1$ and let $R$ be its ring of integers (cf. [[Extension of a field|Extension of a field]]; [[Norm on a field|Norm
5 KB (919 words) - 04:20, 15 February 2024
...lpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $| \alpha | = \alpha _ { 1 } + \ldots + \alpha _ { n }$, $z ^ { \alpha } =
7 KB (996 words) - 09:31, 3 February 2024
...} )$, i.e., $\lambda$ is a weakly decreasing sequence of $l$ non-negative integers $\lambda _ { i }$ called parts. Associated with any partition is an [[alter
.... For the example tableau, $\pi_T = 3111324$. Also, a sequence of positive integers $\pi = w _ { 1 } \dots w _ { n }$ is a lattice permutation or ballot sequen
14 KB (2,001 words) - 10:09, 11 November 2023
are integers, then interchanging in (2) the positions of $ \alpha $
8 KB (1,027 words) - 08:38, 14 July 2022
...> <TD valign="top"> E.L. Post, "Recursively enumerable sets of positive integers and their decision problems" ''Bull. Amer. Math. Soc.'' , '''50''' (1944)
6 KB (862 words) - 19:26, 22 January 2016
...{Z} _ { p }$, where $\mathbf{Z} _ { p }$ is the additive group of $p$-adic integers. Then there is a sequence of fields
...of $A _ { n } ( k )$. Iwasawa [[#References|[a8]]] proved that there exist integers $\lambda _ { p } ( K / k ) \geq 0$, $\mu _ { p } ( K / k ) \geq 0$ and $\nu
19 KB (2,876 words) - 05:38, 15 February 2024
For any integers $ a, b, c $(
7 KB (1,143 words) - 07:00, 6 July 2023
into positive integers $ m _{i} $ ,
6 KB (816 words) - 16:46, 17 December 2019
of integers of the field $ K $,
5 KB (793 words) - 16:10, 1 April 2020
...tribution|Multi-dimensional distribution]]) are defined similarly: For any integers $ k _ {i} \geq 0 $,
7 KB (1,019 words) - 08:53, 21 January 2024
be a sequence of real numbers (indexed by the non-negative integers). A regularization of $ \{ a _ {n} \} $
7 KB (1,067 words) - 12:49, 12 March 2021
...io $\lambda_i/\lambda_j$ is irrational, then for all $\eta\geq0$ there are integers $x_1,\ldots,x_s$, not all zero, such that $\lvert x_1\lambda_1+\cdots+x_s\l
5 KB (868 words) - 21:24, 18 November 2016
...s an instruction labelled by this label. Fundamental data types are lines, integers and real numbers, names, and models. The basic type is the line, and all ot
6 KB (915 words) - 08:14, 6 June 2020
...arbitrary commutative associative ring $R$, for example, over the ring of integers, are also discussed. The accepted notation for the ring of polynomials in a
6 KB (1,011 words) - 06:01, 30 September 2023
...ps is of course the multiplicative semi-group $\mathbf{N}$ of all positive integers $\{ 1,2,3, \ldots \}$, with its subset $P_{\mathbf{N}}$ of all rational pri
...i-group $G_K$ of all non-zero ideals in the ring $R=R(K)$ of all algebraic integers in a given
14 KB (2,037 words) - 09:45, 11 November 2023
are arbitrary non-negative integers. If all terms of the series (1) are non-negative, the sequence of its parti
7 KB (1,082 words) - 19:36, 5 June 2020
each run through all the integers in an independent manner, is known as the lattice (of points) $ \Lambda $
then there exist integers $ x _ {1} \dots x _ {n} $,
28 KB (4,011 words) - 11:37, 19 January 2022
is the maximal ideal of the ring of integers of an algebraic closure of $ K $ .
is the ring of integers of a local field of residue characteristic $ p $
17 KB (2,537 words) - 22:38, 15 December 2019
...be the multiplication operation on real numbers, the addition operation on integers, etc. Suppose that the first of these interpretations is used for $\cdot$.
7 KB (1,091 words) - 18:50, 14 August 2014
Let $n$, $d$ be positive integers with $d \geq 2$ and $\alpha$ an algebraic number with $0 < | \alpha | < 1$.
6 KB (816 words) - 20:26, 5 December 2023
...lume element in $\Omega$. Here, all $q_j$ and $s _ { j }$ are non-negative integers. Note that the integral moments $c _ { q }$ depend on $f$ and $\Gamma$, but
6 KB (978 words) - 17:44, 1 July 2020
are non-negative integers, $ | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} $,
7 KB (986 words) - 14:54, 7 June 2020
are integers. There are positive constants $ c $,
6 KB (922 words) - 06:29, 30 May 2020
...arrow \mathbf{F} _ { p }$, where $\mathbf{F} _ { p }$ denotes the field of integers modulo $p$. Using the [[Dirichlet theorem|Dirichlet theorem]] on prime numb
...ickson polynomials have been obtained for the ring $\mathbf{Z} _ { n }$ of integers modulo $n$ as well as for the Galois ring $\operatorname{GR} ( p ^ { r } ,
15 KB (2,207 words) - 16:45, 1 July 2020
...les (the (disjoint) cycle decomposition of a permutation); the sequence of integers
7 KB (1,003 words) - 08:24, 6 June 2020
Any ring can be regarded as an algebra over the ring of the integers by taking the product $ n a $ (where $ n $
...uirements of the theory of algebraic equations, and residue-class rings of integers in number theory. The general concept of a ring did not exist.
21 KB (3,225 words) - 09:25, 13 July 2022
where $\Z$ is the set of all integers, then
6 KB (1,047 words) - 11:37, 13 February 2024
is the quotient group of the real numbers by the integers, associates to each group its group of characters.
8 KB (1,241 words) - 19:40, 5 June 2020
...king" well-defined because the $\Gamma$ function has poles in all negative integers. However, given the [[Laurent series]] of $\Gamma$ at such poles and the fa
6 KB (1,108 words) - 07:54, 26 March 2023
...a types can be viewed as Herbrand universa. For instance, the non-negative integers $ 0,1,2, \dots $
.... For instance, consider the simple case of addition over the non-negative integers. This operation is determined by the following two recursion equations:
15 KB (2,311 words) - 04:11, 6 June 2020
...ion (cf. [[Diophantine equations]]). In the simplest case of a solution in integers $(x_0,\ldots,x_n)$ of a Diophantine equation, the height is a function of t
6 KB (893 words) - 08:43, 1 May 2023
...uence $x ( n )$, $n \in \mathbf{Z}$, that is identically zero for negative integers, is defined as
9 KB (1,351 words) - 19:32, 2 February 2024
has a solution in integers $ a _ {0} \dots a _ {n - 1 } $,
of integers with condition (2) satisfies condition (1).
8 KB (1,143 words) - 08:03, 6 June 2020
...the first time that $S _ { n } = - J$ and decides $H _ { 0 }$ for positive integers $J$ and $K$. In this case $S _ { 1 } , S _ { 2 } , \ldots$ is a [[Random wa
8 KB (1,155 words) - 19:33, 2 February 2024
where the integers $ h _ {i} , h _ {i} ^ \prime $
7 KB (963 words) - 14:55, 7 June 2020
are coprime integers, and is determined by the rational number $ r = {p / q } $.
7 KB (1,038 words) - 11:33, 26 March 2023
are integers and $ 2 \pi i $
11 KB (1,593 words) - 19:37, 5 June 2020
...the solutions are looked for either in that field or else in its rings of integers. The theory of algebraic functions developed in parallel with the theory of
be a form with rational coefficients, and let there exist a solution in integers $ ( x _ {0} , y _ {0} ) $
24 KB (3,602 words) - 11:48, 26 March 2023
...n $ with coefficients in the ring $ \mathbf Z _{p} $ of $ p $ -adic integers (with the topology induced by that determined by the $ p $ -adic norm of
8 KB (1,117 words) - 20:03, 27 February 2021
is a collection of complex integers (a lattice domain) and that the Green function $ G _ {z} ( z, z ^ \prim
8 KB (1,168 words) - 20:13, 12 January 2024
are called integers (it is assumed that $ +a = a $;
...tural number, are known as rational numbers or fractions. They include all integers. The number $ (a)1 $
26 KB (4,086 words) - 09:51, 4 April 2020
of non-negative integers satisfying $ \lambda _ {1} \geq \dots \geq \lambda _ {n} $
14 KB (1,916 words) - 08:00, 5 May 2022
...ional ruled surfaces $F_n=P(\mathcal O_{P^1}+\mathcal O_{P^1}(n))$ for all integers $n\geq2$ and $n=0$.
7 KB (1,005 words) - 21:13, 21 November 2018
integers $n_1,\dots,n_k$ such that $n_1+2n_2+\cdots +kn_k = n$, and they are related
7 KB (1,115 words) - 16:05, 22 January 2021
...h includes the study of operations performed on numbers. The properties of integers form the subject of number theory (cf. [[Elementary number theory|Elementar
...a of that period was of a high standard. Indian mathematicians operated on integers and fractions using methods very similar to our own. They solved many probl
20 KB (3,091 words) - 18:48, 5 April 2020
theory]], is also an exterior algebra (over the ring of integers).
7 KB (1,013 words) - 19:38, 5 June 2020
are integers, to find a similar expression for $ e $.
8 KB (1,278 words) - 19:39, 5 June 2020
...let $R^*$ be its multiplicative group. $K_0(R) = \Z$ is then the group of integers, $K_1(R) = R^*/[R^*,R^*]$; and $K_2(\Z)$ is the cyclic group of order two.
...ems for the functors $K_i(R)$ and $G_i(R)$. In fact, if $A$ is the ring of integers or the ring of polynomials over a finite field, and $R$ is an $R$-order and
14 KB (2,405 words) - 22:14, 10 January 2015
...><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.K. van Douwen, "The integers and topology" K. Kunen (ed.) J.E. Vaughan (ed.) , ''Handbook of Set-Theor
7 KB (1,075 words) - 19:29, 1 January 2021
is the set of integers.
8 KB (1,242 words) - 07:04, 6 May 2022
...also [[Orthogonal basis|Orthogonal basis]]) for $H$. For any non-negative integers $n _ { 1 } , n _ { 2 } , \dots$ such that $n _ { 1 } + n _ { 2 } + \ldots =
9 KB (1,318 words) - 17:43, 1 July 2020
positive integers if there are no special requirements, and $ n $
7 KB (1,029 words) - 10:58, 29 May 2020
...Namely, let $d \geq 1$ and $\{ h _ { i } \} _ { 0 \leq i \leq d - 1 }$ be integers. Then there exists a Buchsbaum local ring $A$ with $\operatorname { dim } A
...\alpha _ { d } ^ { n _ { d } }$ forms a $d$-sequence in any order for all integers $n _ { i } \geq 1$. Such a sequence is called an unconditioned strong $d$-s
27 KB (4,003 words) - 17:43, 1 July 2020
with the integers $ \{ 1 \dots \rho ( t ) \} $
8 KB (1,232 words) - 08:28, 12 November 2023
...say, assuming the validity of the continuum hypothesis, for any triple of integers $ n $,
9 KB (1,337 words) - 06:29, 30 May 2020
...have become their constituent parts. Problems related to the properties of integers, which subsequently brought about number theory, were typical of this perio
9 KB (1,264 words) - 16:56, 15 April 2012
is the subgroup of the integers (or real numbers modulo 1). In Euclidean space $ X \subset \mathbf R ^
9 KB (1,321 words) - 17:45, 4 June 2020
...solvable elementary theories are those of addition of natural numbers and integers, of Abelian groups, of $ p $-
9 KB (1,436 words) - 19:37, 5 June 2020
...d J. Tannery (1894) constructed a theory of rational numbers from pairs of integers.
The system of integers $ \mathbf Z = \langle Z , + , \cdot , 0 , 1 \rangle $
23 KB (3,482 words) - 08:03, 6 June 2020
are integers. The subgroup $ Q (R) $
and it can be described as the set of algebraic integers of the cyclotomic field generated by a cubic root of unity with norm $ 1
22 KB (3,351 words) - 19:14, 21 December 2019
...verline { m } = \{ m _ { n } \} _ { n = 0 } ^ { \infty }$ be a sequence of integers $\geq 2$, called a branch index. Let $T _ { \overline{m} }$ be a sphericall
8 KB (1,322 words) - 13:08, 10 February 2024
...al segment of the ordinal numbers or as the set of (positive and negative) integers, and also systems in which all formulas are objects of a well-defined type,
8 KB (1,261 words) - 20:30, 13 December 2019
copies of the group of integers $ \mathbf Z $;
are integers. In applications, branched coverings are also of importance. Consider a map
20 KB (3,075 words) - 08:26, 6 June 2020
the ring of integers, a ring of functions, or an arbitrary associative
integers. Then a matrix $A\in \M_{m,n}(K)$ can be written in the form
18 KB (3,377 words) - 17:54, 2 November 2013
has a solution in integers $ x,\ y,\ z $,
9 KB (1,344 words) - 16:08, 6 February 2020
...cyclopediaofmath.org/legacyimages/m/m064/m064430/m0644305.png" /> are real integers (this group is called the [[Modular group|modular group]]). The transformat
39 KB (5,287 words) - 17:07, 7 February 2011
...tation with a highest weight vector]]). Given an $n$-tuple of non-negative integers $\def\L{\Lambda}\def\l{\lambda}\L=(\l_1,\dots,\l_n)$, the integrable highes
..._0$ for $i=1,\dots,r$, where the $a_{0i}, a_{i0}$ are certain non-positive integers, and one puts
15 KB (2,514 words) - 20:27, 15 November 2017
are coprime integers, $ 0 \leq \nu < \mu $).
10 KB (1,350 words) - 12:59, 28 April 2024
are integers.
11 KB (1,603 words) - 16:08, 1 April 2020
of integers, where the multi-index $ i _ {1} \dots i _ {k} $
A set of such integers $ \{ b _ {i _ {1} } \dots b _ {i _ {k} } \} $
42 KB (6,290 words) - 19:33, 17 January 2024
...ion coinciding with $f$ at each point $2 \pi k / N$, where $k , N > 0$ are integers, $k < N$, and $f _ { N }$ is linear on intervals. Suppose $f \in \mathcal{A
10 KB (1,493 words) - 08:15, 15 February 2024
...from the vertex set $V$ of a digraph $G = ( V , E )$ into the non-negative integers ${\bf Z} ^ { 0 }$ defined inductively by: $g = \operatorname { mex } g ( F
13 KB (1,999 words) - 16:52, 18 February 2024
if and only if there are integers $ a , b , c , d \in \mathbf Z $,
10 KB (1,413 words) - 07:59, 6 June 2020
are symmetrically additive for all pairs of integers $ m, n \geq 2 $
9 KB (1,370 words) - 22:15, 5 June 2020
...ww.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642057.png" /> of
integers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o
....encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420242.png" /> and
integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
66 KB (9,085 words) - 17:28, 31 March 2020
and any collection of non-negative integers $ n _ {1} \dots n _ {k} $
11 KB (1,537 words) - 19:39, 5 June 2020
is a chain with the order type of the negative integers [[#References|[2]]].
10 KB (1,515 words) - 08:10, 6 June 2020
..._{1} +m _{3} \omega _{3} $ , and $ m _{1} ,\ m _{3} $ run through all integers except $ m _{1} = m _{3} = 0 $ . The function $ \wp (z) $ is an even el
11 KB (1,535 words) - 13:55, 18 May 2023
...ell as a discrete analogue applicable to distributions concentrated on the integers.
9 KB (1,390 words) - 07:41, 4 February 2024
are non-negative integers (cf. [[Cartan theorem|Cartan theorem]] on the highest weight vector). For a
10 KB (1,344 words) - 08:11, 6 June 2020
is a category the objects of which are all integers plus the null object $ N $,
10 KB (1,515 words) - 18:19, 31 March 2020
...: n=0,1,...\}$, starting at an integer $x$, is a stochastic process on the integers, given by $S_0=x$, $S_n = x +X_1+...+X_n (n\geq1)$, where $X_n, n\geq1$, is
For $c\leq x\leq d$ integers, the probability $\pi(x)$ that a simple random walk, starting at $x$, reach
11 KB (1,725 words) - 19:55, 5 March 2024
The space $W^l_p(\Omega)$ can also be defined for negative integers $l$. Its elements are usually generalized functions, that is, linear functi
8 KB (1,334 words) - 17:47, 30 November 2012
...)$-matrix ($n \geq 2$), which uses Cassini ovals instead of discs. For any integers $i$ and $j$ ($1 \leq i , j \leq n$) with $i \neq j$, the $( i , j )$th Cass
10 KB (1,639 words) - 07:05, 15 February 2024
1) Any Abelian group $M$ is a module over the ring of integers $\ZZ$. For $a \in \ZZ$ and $m \in M$ the product $am$ is defined as the res
...umber field, then other $G$-modules are: the additive group of the ring of integers of $K$, the group of units of $K$, the group of divisors and the
23 KB (3,918 words) - 04:31, 23 July 2018
...= \sum _ { k = 1 } ^ { r } a _ { k } D _ { k }$, where $a_k$ are positive integers and $D _ { k }$ are smooth irreducible divisors such that any subset of the
8 KB (1,253 words) - 15:30, 1 July 2020
...tion numbers, which are real numbers when their geometric counterparts are integers. Examples of such real numbers are the heights of points and subvarieties,
8 KB (1,219 words) - 21:00, 13 July 2020
or through the integers in this interval. In statistical problems, the distribution $ P ^ {T} $
runs through the integers of $ [ 0, T] $
27 KB (3,850 words) - 14:55, 7 June 2020
are positive integers, $ e( \alpha ) $
are negative integers, $ e( \alpha ) $
32 KB (4,602 words) - 04:46, 7 January 2022
for certain integers $ N _ {i} $.
12 KB (1,694 words) - 06:42, 26 March 2023
is spanned for all sufficiently large positive integers $ s $.
9 KB (1,417 words) - 16:09, 1 April 2020
For all positive integers $ n $
13 KB (1,981 words) - 20:02, 17 December 2019
...n matrices $W_+$ and $W_-$, also without poles in $\R$ or at $\infty$, and integers $k_1\le \cdots \le k_m$ such that
9 KB (1,294 words) - 22:47, 28 February 2015
are any given integers such that $ r \geq p $,
10 KB (1,586 words) - 09:40, 26 March 2023
For any function $t ( n )$ from the positive integers to itself, the complexity class $\operatorname {DTIME}[t(n)]$ is the set of
...nstructible function is used. Recall that a function $s$ from the positive integers to itself is space constructible (respectively, time constructible) if and
22 KB (3,250 words) - 17:43, 1 July 2020
are integers; the real axis is invariant with respect to modular fractional-linear mappi
13 KB (1,875 words) - 13:58, 17 March 2023
Here, ${\bf Z}_+$ is the set of non-negative integers; $x _ { i j } ^ { h } \in \mathbf{R} ^ { n _ { 1 } }$, respectively $x _ {
15 KB (2,237 words) - 17:01, 1 July 2020
...varphi } \otimes 1 _ { n } \|$ is finite, where the supremum runs over all integers $n \geq 1$ and $1_n$ is the identity operator from the $C ^ { * }$-algebra
14 KB (2,163 words) - 19:56, 8 February 2024
be either the ring of integers $ \mathbf Z $
which are positive integers $ > 1 $
51 KB (7,267 words) - 07:39, 14 January 2024