A Direct Sum Decomposition of the Integers and a Question of Y. Ito

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.

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“…This odd/even dichotomy between the sets S and 2S implies the next result which is fundamental to our analysis (see also [6]). …”

Section: Lemmamentioning

confidence: 84%