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A congruence relating the class numbers of complex quadratic fields. (English) Zbl 0557.12003
Let $$n$$ denote a positive integer and let $$p_ 1,\ldots,p_ s$$ be $$s$$ $$(\geq 0)$$ distinct primes $$\equiv 1\pmod 4$$ and $$q_{s+1},\ldots,q_ n$$ be $$n-s$$ $$(\geq 0)$$ distinct primes $$\equiv 3\pmod 4$$. Set $$d=(-1)^{n-s} p_ 1\cdots p_ s q_{s+1}\cdots q_ n\equiv 1\pmod 4$$. The integer $$d$$ is the discriminant of the quadratic field $$\mathbb Q(\sqrt{d})$$, which is real or complex according as $$n-s$$ is even or odd. The class number of the quadratic field of discriminant $$D$$ is denoted by $$h(D)$$.
In this paper a congruence of the type
\begin{aligned} &\sum_{e| d,\;e>0,\;e\equiv 1\pmod 4}(c_ 1(d,e) h(-4e)+c_ 2(d,e) h(-8e))\\ &+\sum_{e| d,\;e<0,\;e\equiv 1\pmod 4}(c_ 3(d,e) h(e)+c_ 4(d,e) h(8e))\equiv c(d)\pmod{2^{n+2}}\end{aligned} is proved, where $$c_ i(d,e)$$ $$(i=1,\ldots,4)$$ and $$c(d)$$ are certain integers. When $$n=1$$ this congruence reduces to well-known congruences modulo 8 due to Dirichlet and when $$n=2$$ it gives a unified congruence for the 18 congruences modulo 16 proved by A. Pizer [J. Number Theory 8, 184–192 (1976; Zbl 0329.12003)].

##### MSC:
 11R11 Quadratic extensions 11R29 Class numbers, class groups, discriminants