Concluding remarks.

There have been numerous papers over the last few years that have been concerned with sequences of integers for which a `Kummer--type' Theorem, a `Lucas--type' Theorem and/or a `Wolstenholme--type' Theorem holds. One nice example is the Apéry numbers,

which were introduced in Apéry's proof of the irrationality of . At first, a few seemingly surprising congruences were found for these numbers, but in 1982, Gessel showed that these were all consequences of the fact that the Apéry numbers satisfy `Lucas--type' and `Wolstenholme--type' Theorems (that is and for all and primes ). R. McIntosh has asked whether a non--trivial sequence of integers, satisfying a `Lucas--type' Theorem, can grow slower than ?

One can also generalize the notion of binomial coefficients, as follows, and obtain `Kummer--type' and `Lucas--type' Theorems: Given a sequence of integers, define and , and ask what power of a prime p divides , and also for the value of . The first of these questions is attacked systematically in a beautiful paper of Knuth and Wilf. A nice example was given by Fray, who proved `Kummer--type' and `Lucas--type' Theorems for the sequence of `q--binomial coefficients' (where each ).

There are a number of questions that have recieved a lot of attention in the literature which do not concern us here. Many require straightforward manipulations of some of the results given here (for instance, how many entries of a given row of Pascal's triangle are not divisible by p), others easy generalizations (for instance to multinomial coefficients --- most results in that area follow immediately from the fact that multinomial coefficients can be expressed as a product of binomial coefficients). People have also investigated the density of entries in Pascal's triangle divisible by any given integer n, and strong estimates of the average (and various connections therein to fractals and cellular automata). For these questions, and some others that are not covered here, the reader should look at Boyd, Cook and Morton, Singmaster and Stolarsky.