Universally bad integers and the 2-adics

Abstract

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair ða; bÞ of positive odd integers good, if Z ¼ aS~2bS; where S is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair ða; bÞ bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if ðua; bÞ is bad for all pairs of positive odd integers a and b: De Bruijn showed that all positive integers of the form u ¼ 2 k þ 1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u ¼ f p k ð4Þ are universally bad, where p is prime and f n ðxÞ is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u ¼ f p k ð4Þ are in fact de Bruijn universally bad for all primes p42: Furthermore, we show that the universally bad integers f 2 k ð4Þ; and more generally, those of the form 4 k þ 1; are not de Bruijn universally bad. r 2004 Elsevier Inc. All rights reserved.

Additive systems and a theorem of de Bruijn

“…There are many interesting recent results about additive systems for Z, for example, [1,5,6,7,10,17,19]. However, de Bruijn's remark at the end of his 1956 paper on N 0 still accurately describes the current state of the problem: "Some years ago the author [3] discussed various aspects of the analogous problem for number systems representing uniquely all integers (without restriction to nonnegative ones).…”

De combinatoriek van De Bruijn

“…In 1950 verschijnt een artikel [11] wat een inspiratiebron wordt voor veel later werk, van zowel De Bruijn zelf als van veel anderen, eg, [38,39]. Als voorbeeld noem ik een concept waar De Bruijn zijn naam aan verleent; de 'Moser-De Bruijn rij' [17,61].…”