演講訊息 TALKS

The studies on the congruence of combinatorial numbers can be traced back to the famous and also age-old Pascal’s Fractal formed by the triangle of parities (0 and 1 digits) of binomial coefficients . As a pioneer of this area, Kummer formulated the maximum power of dividing . Lucas, another pioneer, developed a very useful tool that , where is a prime, “ ” denotes congruence modulo , and denotes the sequence of digits representing in base . The congruence of combinatorial numbers modulo was considered for a long time, but it is hard and just few results came out. In the first part of this talk, I will present the congruences of Catalan numbers and Motzkin numbers modulo 4 and 8. In particular, this work justifies the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. The units of the modulo ring are the numbers coprime (relatively prime) to . The set of all units forms the multiplicative group of integers modulo n, usually denoted by . It is well known that is isomorphic to , and the group is cyclic for any odd prime . In the second part of this talk, I will present a newest work. By transforming the multiplicative group to its twin as an additive group, we can develop efficient and powerful formulas for enumerating the congruences of many combinatoric numbers, including new proofs for Kummer’s and Lucas’ Theorems.