Universally bad integers and the 2 - adics

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair ða; bÞ of positive odd integers good, if Z 1⁄4 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 1⁄4 2 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 1⁄4 fpk ð4Þ are universally bad, where p is prime and fnðxÞ 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 1⁄4 fpk ð4Þ are in fact de Bruijn universally bad for all primes p42: Furthermore, we show that the universally bad integers f2k ð4Þ; and more generally, those of the form 4 þ 1; are not de Bruijn universally bad. r 2004 Elsevier Inc. All rights reserved.