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We give a new proof of a recent theorem of Kaplansky and use it to revive an old, seemingly forgotten result of Glaisher. It is well known that, for any n > 0, the prime numbers of the form x +ny can be described by congruence conditions if and only if n is one of Euler’s convenient numbers [4, p. 62]. Since 1, 2, 4, 8 and 16 are the only convenient powers of 2, the following theorem of Kaplansky [8] is remarkable: A prime p ≡ 1 (mod 16) is representable by both or none of x + 32y and x + 64y, whereas a prime p ≡ 9 (mod 16) is representable by exactly one of these forms. Kaplansky writes, “Although this is a simple elementary statement I do not have a direct proof. Instead I shall show that it is a quick corollary of two significant theorems”. The latter are reciprocity laws concerning 2 and –4 as fourth and eighth power residues. In this note we give a new proof of Kaplansky’s theorem that aspires to be direct. Instead of reciprocity it uses an idea of Aigner [1], Barrucand and Cohn [2], namely that both representations p = u + v and p = z + 2w of a prime p ≡ 1 mod 8 come from a single representation of p by the norm form of the eighth cyclotomic field. Consider an odd prime p and let h and h′ be the class numbers corresponding to the discriminants −4p and −8p, respectively. Using Kaplansky’s theorem, we give a quick proof of an old, seemingly forgotten result of Glaisher: If p ≡ 1 (mod 16), then either both or none of h and h′ are divisible by 8; if p ≡ 9 (mod 16), then exactly one of these class numbers is divisible by 8. This was originally demonstrated in [5, §12] along with other interesting class number relations using Dirichlet’s class number formula. Proof of Kaplansky’s theorem. Consider a prime p ≡ 1 (mod 8). It splits in the eighth cyclotomic field Q(ζ), ζ + 1 = 0. Therefore, and since the ring 1 James W. L. Glaisher (1848–1928), son of the meteorologist and world record holding balloonist of the same name.

Two theorems of Glaisher and Kaplansky