Abstract. In 1960, Sierpiński proved that there exist infinitely many odd positive rational integers k such that is composite in for all . Any such integer k is now known as a Sierpiński number, and the smallest value of k produced by Sierpiński's proof is . In 1962, John Selfridge showed that is also a Sierpiński number, and he conjectured that this value of k is the smallest Sierpiński number. This conjecture, however, is still unresolved today. In this article, we investigate the analogous problem in the ring of integers of each imaginary field having class number one. More precisely, for each , with , that has unique factorization, we determine all , with minimal odd norm larger than 1, such that is composite in for all . We call these numbers Selfridge numbers in honor of John Selfridge.