Davis, Kenneth S.; Webb, William A. Lucas’ theorem for prime powers. (English) Zbl 0704.11002 Eur. J. Comb. 11, No. 3, 229-233 (1990). Lucas’ theorem on binomial coefficients states that \(\left( \begin{matrix} A\\ B\end{matrix} \right)\equiv \left( \begin{matrix} a_ r\\ b_ r\end{matrix} \right)...\left( \begin{matrix} a_ 1\\ b_ 1\end{matrix} \right)\left( \begin{matrix} a_ 0\\ b_ 0\end{matrix} \right)(mod p)\), where p is a prime and \(A=a_ rp^ r+...+a_ 1p+a_ 0\), \(B=b_ rp^ r+...+b_ 1p+b_ 0\) are the p-adic expansions of A and B. The authors show that a similar formula holds modulo \(p^ s\) with \(s\geq 2\) where the product involves a slightly modified binomial coefficient evaluated on blocks of s digits. Reviewer: P.Kiss Cited in 3 ReviewsCited in 18 Documents MSC: 11A07 Congruences; primitive roots; residue systems 11B65 Binomial coefficients; factorials; \(q\)-identities 05A10 Factorials, binomial coefficients, combinatorial functions Keywords:congruence; binomial coefficients PDFBibTeX XMLCite \textit{K. S. Davis} and \textit{W. A. Webb}, Eur. J. Comb. 11, No. 3, 229--233 (1990; Zbl 0704.11002) Full Text: DOI Online Encyclopedia of Integer Sequences: Triangle, read by rows, formed by reading Pascal’s triangle (A007318) mod 4. References: [1] Dickson, L. E., History of the theory of numbers, Vol. 1, Chelsea, New York (1952) [2] Fine, N. J., Binomial coefficients modulo a prime, Am. Math. Monthly, 54, 589-592 (1947) · Zbl 0030.11102 [3] Kazandzidis, G. S., Congruences on binomial coefficients, Bull. Soc. Math. Grièce (NS), 9, 1-12 (1968) · Zbl 0179.06601 [4] D. Singmaster, Divisibility of binomial and multinomial coefficients by primes and prime powers, From a collection of manuscripts related to the Fibonacci Sequence.; D. Singmaster, Divisibility of binomial and multinomial coefficients by primes and prime powers, From a collection of manuscripts related to the Fibonacci Sequence. · Zbl 0517.10013 [5] Singmaster, D., Notes on binomial coefficients I—a generalization of Lucas’ congruence, J. Lond. Math. Soc., 8, 2, 545-548 (1974) · Zbl 0293.05005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.