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Some congruences for binomial coefficients. (English) Zbl 1070.11007
Miyake, Katsuya (ed.), Class field theory – its centenary and prospect. Proceedings of the 7th MSJ International Research Institute of the Mathematical Society of Japan, Tokyo, Japan, June 3–12, 1998. Tokyo: Mathematical Society of Japan (ISBN 4-931469-11-6/hbk). Adv. Stud. Pure Math. 30, 445-461 (2001).
Summary: Suppose that $$p=tn+r$$ is a prime and that $$h$$ is the class number of the imaginary quadratic field, $$\mathbb{Q}(\sqrt{-t})$$. If $$t \equiv 3\pmod 4$$ is a prime, just $$r$$ is a quadratic residue modulo $$t$$ and the order of $$r$$ modulo $$t$$ is $$\frac{t-1}{2}$$ then $$4p^h$$ can be written in the form $$a^2+tb^2$$ for some integers $$a$$ and $$b$$. And if $$t=4k$$ where $$k\equiv 1\pmod 4$$, $$r\equiv 3\pmod 4$$, $$r$$ is a quadratic non-residue modulo $$t$$ and the order of $$r$$ modulo $$t$$ is $$k-1$$, then $$p^h=a^2+kb^2$$ for some integers $$a$$ and $$b$$. Our result is that $$a$$ or $$2a$$ is congruent modulo $$p$$ to a product of certain binomial coefficients modulo sign. As an example, we give explicit formulas for $$t=11,19,20$$ and 23.
For the entire collection see [Zbl 0968.00031].

##### MSC:
 11B65 Binomial coefficients; factorials; $$q$$-identities 11A07 Congruences; primitive roots; residue systems 11R18 Cyclotomic extensions 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 11T24 Other character sums and Gauss sums