Arithmetic Properties of Binomial Coefficients (Organic Mathematics Proceedings)
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|Many great mathematicians of the nineteenth century considered problems involving binomial coefficients modulo a prime power (for instance Babbage, Cauchy, Cayley, Gauss, Hensel, Hermite, Kummer, Legendre, Lucas and Stickelberger). They discovered a variety of elegant and surprising Theorems which are often easy to prove. This article exhibits most of these results, and extends them in a variety of ways. Chapters on: elementary number theory and the proof of Theorem 1; binomial coefficients modulo prime powers; recognizing the primes; Pascal's triangle via cellular automata; studying binomial coefficients through their generating function; sums of binomial coefficients; Bernoulli numbers and polynomials; generalization of Morley's Theorem; some useful p--adic numbers; and congruences modulo powers of primes. Another version of the article is available at Granville's site.|
|Math Topics:||Prime Numbers, Cellular Automata, Number Theory|
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