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Many of the great mathematicians of the nineteenth century considered problems involving binomial coefficients modulo a prime power (for instance Babbage, Cauchy, Cayley, Gauss, Hensel, Hermite, Kummer, Legendre, Lucas and Stickelberger --- see [Dickson]). They discovered a variety of elegant and surprising Theorems which are often easy to prove. In this article we shall exhibit many of their results, and explore several extensions.

In 1852 Kummer showed that the power of prime **p** which divides the binomial
coefficient is given by the number of `carries' when we
add **m** and **n-m** in base **p**.

In 1878 Lucas gave a method to easily determine
the value of : Let and be the least
non--negative residues of **m** and , respectively. Then

Note that if **p** divides then (1.1) follows easily from Kummer's
Theorem. However, if is the exact power of **p** dividing ,
then we might ask for the value of .
This is given by a result discovered by each of Anton (1869),
Stickelberger (1890) and Hensel (1902) (and many others since!), which
shows that

** Theorem 1.** * Suppose that prime power and positive
integers n=m+r are given. Write in base p,
and let be the least positive residue of
for each (so that ).
Also make the corresponding definitions for .
Let denote the number of `carries', when adding m and r in base p,
on or beyond the jth digit (note that here is the same as k above). Then
*

Taking **q=1** in (1.3) gives (1.2). Note that (1.3) may be re--written in terms of
factorials, since each .
A similar result was recently given by Davis and Webb [4].

Theorem 1 provides a quick way to compute the value of binomial coefficients modulo arbitrary prime powers. In fact we will show that this takes just elementary operations, in section 3.

Wilson's Theorem (which actually appears in earlier work of Liebnitz)
states that for all primes
**p**. An easy consequence of this is that
for all integers **n**.
In 1819 Babbage noticed that, further,
for all primes , and
Wolstenholme, in 1862, that

Another generalization of Wolstenholme's congruence is given by

** Theorem 2.** * Suppose that prime p and positive integers
u and r are given. Then
*

** Remark:** In fact in Theorem 2 only when **p=2** and
is odd.

Note how Theorem 2 allows us to express any in terms of with . Using this result and Theorem 3 below, we will see how to compute factorials very rapidly, in section 3.

In 1899 Glaisher observed that the number of odd entries in any given
row of Pascal's Triangle is a power of **2**. This follows from Kummer's
Theorem by noting that is odd if and only if there
are no carries when adding **m** and **n-m** in base 2; in other words that
the digits `1' in the binary expansion of **m** are a subset of those in the
binary expansion of **n**. Clearly if there are **k** digits `1' in the
binary expansion of **n**, then there are possible subsets of these
`1's, and each corresponds to a value of **m** --- thus there are
odd entries in the **n**th row of Pascal's Triangle.

Larry Roberts also has an elegant (unpublished) result, depending on
Kummer's Theorem: Let be the binary number whose **m**th digit
is modulo **2**; in other words, the integer formed
by reading the **n**th row of Pascal's Triangle, modulo **2**, from left to right.
Then , where the sum is over those values of **m**, for
which the digits `1' in its binary expansion are a subset of those of **n**.
Thus if (where )
then

In section 5 we give somewhat different proofs of these last two results, and of Lucas' Theorem, using cellular automata.

In a recent paper [8], using methods from both elementary number theory
and the theory of cellular automata, we extended this result of
Glaisher's to the entries in Pascal's triangle that are :
specifically we showed that in any given row of Pascal's triangle,
the number of entries that are congruent to is either
**0** or a power of **2**. Similarly for .
We then extended this to and .
The likelihood of a general result of this type begins to emerge, and
to find out more the reader is encouraged to look at [8].

As is so well known, for any fixed **n**, the sum, over all **m**,
of the binomial coefficients , is exactly
2 to the power **n**. What is less well-known is that the
sum of the binomial coefficients, over all **m** in certain
fixed residue classes, sometimes satisfy certain surprising
congruences:

In 1876 Hermite showed that if **n** is odd then the sum of the binomial
coefficients , over those positive integers **m** that are
divisible by **p-1**, is divisible by **p**.

In 1899 Glaisher generalized this by showing that for
any given prime **p** and integers , we have

In 1953 Carlitz generalized Hermite's Theorem to prime
powers: If divides **n**, with and , then

In 1913 Fleck gave the related result that for
any given prime **p** and integers , we have

In 1965 Bhaskaran showed that if **p** is an odd prime then **p+1** divides **n**
if and only if

In 1895 Morley [18] showed that for any prime ,

His ingenious proof, which is based on an explicit form of De Moivre's Theorem, can be modified to show that (1.14) holds modulo if and only if
There are many results in the literature that relate the value
of binomial coefficients of the form
, for a given, fixed **d>0**
dividing **p-1**, to representations of the prime **p** by certain quadratic
forms (see [10]). The first such result, due to Gauss (1828), is for **d=4**: \
Write any prime as , and choose
the sign of **a** so that ; then
.
Recently, Beukers conjectured that

* Many thanks to the numerous mathematicians who have
sent me their comments, results and offprints to include in this survey,
particularly Neil Calkin, Fred Howard, Larry Roberts, Herb Wilf and Kenneth Williams.*

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