Representing integers as linear combinations of power products
نویسندگان
چکیده
منابع مشابه
Representing integers as linear combinations of powers
At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. (See his paper in the present volume.) He proposed some related problems on the representation of integers by sums or differences of powers of 2 and of 3. In this note we prove some results on this problem and the more general problem about the representation by line...
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In this paper we consider representations of algebraic integers of a number field as linear combinations of units with coefficients coming from a fixed small set, and as sums of elements having small norms in absolute value. These theorems can be viewed as results concerning a generalization of the so-called unit sum number problem, as well. Beside these, extending previous related results we g...
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Let k-l,ml,..~+k denote non-negative integers, and suppose the greatest common divisor of ml,...,mk is 1 . We show that if '1, "',sk are sufficiently long blocks of consecutive integers, then the set mlSl+ . ..+mkSk contains a sizable block of consecutive integers. For example; if m and n are relatively prime natural numbers, and U, U ? Vt V are integers with U-u 2 n-l , V-v 2 m-l 1 then the se...
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We study in detail the special case of Waring’s problem when the power k = 2. Ultimately, we prove that four is the least number of squares needed to represent any integer. To this end, we prove that some numbers cannot be represented as sums of two squares, some cannot be represented as sums of three, and all can be represented as sums of four. We also show that numbers of a certain form can b...
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1. Throughout this paper, c 1 , c2 , . . . denote absolute constants ; ko (a, fl, . . .), kr (a, f3, . . .), . . ., xo (a, /3, . . .), . . . denote constants depending only on the parameters a, /l, . . . ; v(n) denotes the number of the prime factors of the positive integer n, counted according to their multiplicity . The number of the elements of a finite set S is denoted by I S I . Let k, n b...
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ژورنال
عنوان ژورنال: Archiv der Mathematik
سال: 2012
ISSN: 0003-889X,1420-8938
DOI: 10.1007/s00013-012-0380-4