منابع مشابه
On finite sets which tile the integers
A set of integers A is said to tile the integers if there is a set C ⊂ Z such that every integer n can be written in a unique way as n = a + c with a ∈ A and c ∈ C. Throughout this paper we will assume that A is finite. It is well known (see [7]) that any tiling of Z by a finite set A must be periodic: C = B + MZ for some finite set B ⊂ Z such that |A| |B| = M . W then write A ⊕ B = Z/MZ. Newma...
متن کاملThe numerical semigroup of the integers which are bounded by a submonoid of N2
Let N be the set of nonnegative integers. A submonoid of (N,+) is a subset M of N that is closed for the addition and that contains the zero element. A numerical semigroup is a submonoid S of (N,+) such that N \ S is finite. Definition 1 Let S be a numerical semigroup. The Frobenius number of S is F(S ) = max(Z \ S ) and the genus of S is g(S ) = (N \ S ). Definition 2 A Frobenius variety is a ...
متن کاملOn integers which are the sum of a power of 2 and a polynomial value
Here, we show that if f(x) ∈ Z[x] has degree at least 2 then the set of integers which are of the form 2k+f(m) for some integers k ≥ 0 and m is of asymptotic density 0. We also make some conjectures and prove some results about integers not of the form |2k ±ma(m− 1)|.
متن کاملSeveral Results on Sequences Which Are Similar to the Positive Integers
Sequence of positive integers {xn}n≥1 is called similar to N with respect to a given property A if for every n ≥ 1 the numbers xn and n are in the same class of equivalence with respect to A (xn ∼ n(prop A)). If x1 = a(> 1) ∼ 1(prop A) and xn > xn−1 with the condition that xn is the nearest to xn−1 number such that xn ∼ n(prop A), then the sequence {xn} is called minimal recursive with the firs...
متن کاملOn Sets of Integers Which Are Both Sum-free and Product-free
We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1952
ISSN: 0002-9939
DOI: 10.2307/2032272