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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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