...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