A Continued Fractions Approach to a Result of Feit

1. INTRODUCTION. For primes that can be written as a sum of integer squares, p = a 2 + (2b) 2 , Kaplansky [4] asked whether the binary quadratic form F = x 2 − py 2 always represents a and 4b (that is, are there integer solutions to x 2 − py 2 = a and x 2 − py 2 = 4b). Feit [1] and Mollin [4] proved that F does always represent a and 4b using the theory of ideals and the class group structure of quadratic orders. In this Monthly, Walsh [7] proved a more general result using only elementary methods. He showed that if D > 1 is a non-square odd integer, D = a 2 + (2b) 2 , and x 2 − Dy 2 represents −1, then there is a factorization of D into positive integers r and s so that rx 2 − sy