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# Concluding remarks.

There have been numerous papers over the last few years that have been concerned with sequences of integers for which `Kummer-type' Theorems, or `Lucas-type' Theorems or `Wolstenholme-type' Theorems hold. 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 [7] 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 ? Given a sequence of integers, one can generalize the notion of factorials and binomial coefficients by taking 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 [12]. A nice example was given by Fray [6], 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 received 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 various connections therein to fractals and cellular automata). For these questions, and some others that are not covered here, the reader should look at [2], [19] and [20].

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