Universally bad integers and the 2-adics

Abstract

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair ( a , b ) of positive odd integers good, if Z =a S ⊖2b S , where S 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 =2 k +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 = φ p k (4) are universally bad, where p is prime and φ n ( x ) is the n th 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 = φ p k (4) are in fact de Bruijn universally bad for all primes p >2. Furthermore, we show that the universally bad integers φ 2 k (4), and more generally, those of the form 4 k +1, are not de Bruijn universally bad.