Two theorems of Glaisher and Kaplansky
The author presents a proof of the following theorem of \textit{I. Kaplansky} [Proc. Am. Math. Soc. 131, No. 7, 2299--2300 (2003; Zbl 1055.11026)]: A prime \(p \equiv 1\pmod{16}\) is representable by both or neither of \(x^2+32y^2\) and \(x^2+64y^2\), whereas a prime \(p \equiv 9\pmod{16}\) is representable by exactly one of these forms. The proof described here is based on the arithmetic of the eighth cyclotomic field. Also, the relationship of this result to a theorem of \textit{J. W. L. Gaisher} [Q. J. Math. 34, 178--204 (1902; JFM 34.0243.01)] is pointed out.
scientific article; zbMATH DE number 5654316