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Lucas’ theorem for prime powers. (English) Zbl 0704.11002

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

MSC:

11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
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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
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