Kummer's Congruences
Kummer's Congruences are a set of results in number theory that relate to the divisibility of binomial coefficients by prime powers. They were introduced by the mathematician Heinrich Kummer in the 19th century. Specifically, Kummer's work shows how the highest power of a prime that divides a binomial coefficient can be determined using the base-p representations of the integers involved.
These congruences provide insight into the structure of p-adic numbers and have implications in various areas of mathematics, including algebraic number theory and combinatorics. Kummer's findings help mathematicians understand how primes interact with combinatorial objects, enhancing the study of divisibility properties.