On the congruences for the class numbers of the quadratic fields whose discriminants are divisible by 8
Let \(m>0\) be a squarefree odd rational integer, having no prime factor \(\equiv 3\) (mod 4). Let \(\epsilon =T+U \sqrt{2m}>1\) be the fundamental unit of the real quadratic field \({\mathbb{Q}}(\sqrt{2m})\). The author proves the following congruence \[ (1)\quad h(-2m)\equiv (\frac{2}{m}) TU h(2m)\quad (mod 2^{r+2}), \] where r is the number of primes dividing m. Unfortunately the author does not indicate in his paper how (1) is related to congruences already in the literature. We just note that the case \(m=p\) (prime) \(\equiv 5\) (mod 8) of (1) is due to the reviewer [Rocky Mt. J. Math. 11, 19-26 (1981; Zbl 0437.12006)] and that stronger congruences than (1) in the case \(m=p\) (prime) \(\equiv 1\) (mod 4) are due to \textit{P. Kaplan} and the reviewer [Acta Arith. 40, 289-296 (1982; Zbl 0491.12003); ibid. 40, 375-397 (1982; Zbl 0513.12007)].
scientific article; zbMATH DE number 3941639