REMARKS ON THE DIVISIBILITY OF THE CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
AbstractWe consider the divisibility of the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$, where q is an odd prime number, k and n are positive integers. Suppose that k ≡ 1 mod 2 or n ≢ 3 mod 6. We show that the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-3})$ are divisible by n for q ≡ 3 mod 8. This is a generalization of the result of Kishi for imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - 3^n})$ when k ≡ 1 mod 2 or n ≢ 3 mod 6. We also show that the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-1})$ are divisible by n for q ≡ 1 mod 4 and the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-3})$ are divisible by n for q ≡ 7 mod 8.
