Universally bad integers and the 2-adics |
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Authors: | SJ Eigen VS Prasad |
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Institution: | a Northeastern University, Boston, MA 02115, USA b Tokai University, Tokyo, Japan c Department of Mathematical Sciences, University Massachusetts Lowell, Room 428-Q, Olney Science Center, One University Avenue, Lowell, MA 01854-5009, USA |
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Abstract: | In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair (a,b) of positive odd integers good, if , where is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair (a,b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if (ua,b) is bad for all pairs of positive odd integers a and b. De Bruijn showed that all positive integers of the form u=2k+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=φpk(4) are universally bad, where p is prime and φn(x) 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=φpk(4) are in fact de Bruijn universally bad for all primes p>2. Furthermore, we show that the universally bad integers φ2k(4), and more generally, those of the form 4k+1, are not de Bruijn universally bad. |
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Keywords: | Cyclotomic polynomials 2-adic integers Bases for integers Tiling the integers Universally bad integers |
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