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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].

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