ON A QUESTION OF KAPLANSKY II

Abstract

There is a question attributed to Irving Kaplansky concerning the solvability of the quadratic equation $x^2-p{y}^2=a$ in the case that the prime $p=a^2+{\left(2b\right)}^2$. This question was answered in the affirmative by Mollin [1], although according to [3], this result is implicit in the work of Gauss and Legendre. The proof appearing in [1] was later simplified in [4], and it was also shown therein that Kaplansky’s question was a special case of a more general result. Using the method of proof in [4], Mollin [2] has recently extended the results of [4], but upon further consideration, it appears that there is a more general phenomenon occurring, and also, that one of the assumptions in the main theorem of [2] is unnecessary. In this paper we prove this generalization, and eliminate one of the assumptions stated in the main result of [2]. The proof is again based on the method described in [4].