In 1878 Lucas gave a method to easily determine the value of : Let and be the least non--negative residues of m and , respectively. Thenwhere, as usual, denotes the largest integer , and we use the convention if r<s. Re--writing and in base p (so that for each i), this may also be expressed as We will give three very different proofs of Lucas' Theorem: the first, via number theory, in section 2; the second, via cellular automata, in section 5; and the third, via the combinatorics of power series, in section 6.
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 thatwhere r = n-m. Numerous authors have asked whether there is an analogous formula, modulo , for arbitrary . In section 2 we show the following: For a given integer n define to be the product of those integers that are not divisible by p.
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 where is except if p=2 and .
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 .
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, thatfor all primes . In 1952 Ljunggren generalized this to and Jacobsthal to for any integers n>m>0 and prime , where q is the power of p dividing (this exponent q can only be increased if p divides , the rd Bernoulli number). These results, as well as many other similar congruences with larger exponents q, follow easily from Proposition 5 below. For example, if prime then Ljunggren's result above may be re--written as for any integer and . Proposition 5 implies the generalization unless , or 2r+1 = p or , when the congruence holds .
Another generalization of Wolstenholme's congruence is given by
Theorem 2. Suppose that prime p and positive integers u and r are given. Then except perhaps if or 2r+1 = p or , whence the congruence certainly holds , where `' can only be `-' if p=2 (which is then easily determined by evaluating both sides of (1.8) modulo 4), and the integer
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 nth row of Pascal's Triangle.
Larry Roberts also has an elegant (unpublished) result, depending on Kummer's Theorem: Let be the binary number whose mth digit is modulo 2; in other words, the integer formed by reading the nth 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 ) thenwhere is the ith Fermat number. As is well known, all odd constructible numbers are of this form; and thus can be found by reading off the rows of Pascal's Triangle in base 2.
In section 5 we give somewhat different proofs of these last two results, and of Lucas' Theorem, using cellular automata.
In a recent paper , 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 .
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 havefor all positive integers . We prove this in section 6.
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 havewhere .
In 1965 Bhaskaran showed that if p is an odd prime then p+1 divides n if and only iffor . We present proofs of these two results in section 7.
In 1895 Morley  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 p divides ; however it cannot be modified to investigate other binomial coefficients, and so we use a different method for this in section 9. We begin by showing, for , for any . In fact, this product is whenever the p-2nd Bernoulli polynomial vanishes at (which is immediate for m=2).
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 ). 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 thatand this was proved in . In 1846, Jacobi showed that if we write any prime as , where the sign of A is chosen so that , then and this has now been shown to be These congruences, modulo , have only been discovered quite recently (in ) and there are presumably many others waiting to be found.
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.