Universally bad integers and the 2-adics

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair <f>(a,b)</f> of positive odd integers good, if <f>Z=aS⊖2bS</f>, where <f>S</f> is the set of nonnegative integers whose 4-adic expansion has only 0''s and 1''s, otherwise he called the pair <f>(a,b)</f> bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer <f>u</f> is universally bad if <f>(ua,b)</f> is bad for all pairs of positive odd integers <f>a</f> and <f>b</f>. De Bruijn showed that all positive integers of the form <f>u=2k+1</f> 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 <f>u=φpk(4)</f> are universally bad, where <f>p</f> is prime and <f>φn(x)</f> is the <f>n</f>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 <f>u=φpk(4)</f> are in fact de Bruijn universally bad for all primes <f>p>2</f>. Furthermore, we show that the universally bad integers <f>φ2k(4)</f>, and more generally, those of the form <f>4k+1</f>, are not de Bruijn universally bad. [Copyright &y& Elsevier]