Arithmetic properties of Binomial Coefficients

Binomial Coefficients have many remarkable arithmetic properties. In this e-survey we introduce and explain some of what makes binomial coefficients so fascinating. This e-survey is `dynamic' so that it can be edited as soon as new developments occur: if you know of something that you believe should be included please let us know.

Table of Contents

• Introduction.
• Elementary Number Theory.
• Generalization of Lucas' Theorem.
• Congruences for sums of binomial coefficients.
• Computing binomial coefficients modulo prime powers.
• Recognizing the primes.
• Pascal's triangle via cellular automata.
• Studying binomial coefficients through their generating function.
• Bernoulli numbers and polynomials.
• Theorems of Morley and Emma Lehmer and their generalizations.
• Some useful p-adic numbers.
• Congruences modulo higher powers of primes.
• Concluding remarks.
• References.