...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
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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 $f(p^a) = a f(p)$.
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
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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} $
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.
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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
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