...fined on the set of all integers $t=0,\pm1,\pm2,\dots,$ or on the positive integers $t=1,2,\dots$.
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The ring of Gaussian integers or [[Gauss number]]s, $\mathbb{Z}[i]$.
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...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
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...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
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...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 $f(p^a) = a f(p)$.
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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 $k$.
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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\
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...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
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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)
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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} $
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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
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...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
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...r programming]]) but for which some of the variables are constrained to be integers.
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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.
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...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
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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 $b$
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...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
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...al linear transference theorems concern the relations between solutions in integers of a system of homogeneous linear inequalities having a non-singular square
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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 $
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...he above example makes it possible to consider the set of all non-negative integers — the [[Natural sequence|natural sequence]] — as a mathematical object.
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...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
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...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}}</
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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)
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...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
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...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
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...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:
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...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
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...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
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...the language of arithmetic contains one type, namely for the non-negative integers.
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Any set of $m$ integers that are incongruent $\bmod\,m$. Usually, as a complete residue system $\bm
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...e unique integral object in the category of Abelian groups is the group of integers.
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...$p$ is [[Isomorphism|isomorphic]] to the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p$, often denoted $\mathbb{F}_p$ or $\mathrm{GF}(p)$.
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The set of positive integers $ a _ {0} $
of all positive integers $ a _ {0} $
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...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
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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 $(
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* P.J. Cameron and P. Erdős, ''On the number of sets of integers with various properties'', Number theory (Banff, 1988), de Gruyter, Berlin
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...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
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Integers without common (prime) divisors. The [[greatest common divisor]] of two cop
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...lopediaofmath.org/legacyimages/b/b015/b015220/b01522010.png" /> of natural
integers, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o
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...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
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...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 $n$, $n=1,2,\dots$).
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where the minimum is taken over all possible sets of integers $a_1,\ldots,a_n$, not all zero, satisfying the conditions
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are integers and if $ k \neq 0 $
If there exists integers $ n $
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...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
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...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
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...] 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
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...rly any field of characteristic zero is a binomial ring, as is the ring of integers $\mathbf{Z}$.
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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
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...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
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is the ring of integers of $ K $.
be a prime ideal of (the ring of integers $ A _ {K} $)
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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] $.
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...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
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...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
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...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
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...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.'' , '
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...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
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given on this space and taking values in the positive integers). The term "integral automorphism" is mostly used in the non-Soviet liter
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are integers are also periods of $ f ( z) $.
are integers. Analytic functions of one complex variable with more than two primitive pe
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...ma_0,\Gamma_1,\ldots\}$ 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
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...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
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...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
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are not integers, then equation (1) has a fundamental system of solutions of the form
are not integers, where the $ \lambda _ {j} $
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...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]].
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...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
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...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.
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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).
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...on a skew-field the group of values of which is isomorphic to the group of integers $ \mathbf Z $.
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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
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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
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...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
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...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
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...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
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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 $n$
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...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
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...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
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where $N$ is a system of sets of positive integers, called the base of the $\delta$-$\sigma$-operation. See [[Descriptive set
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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
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