Many 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 most of these results, and extend them in a variety of ways. We start with a discussion of what this article contains:
In 1852 Kummer showed that the power of prime p that 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
where, 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: via number theory, via cellular automata, and via the combinatorics of power series.Note that if p divides then (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
where 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
m=n+r are given. Let be the least positive residue of
for each (and
make the corresponding definitions for m and r).
Let be the number of `carries', when adding m and r in base p,
on or beyond the jth digit). Then
where is except if p=2 and .
where is except if p=2 and .
Taking q=1 in (3) gives (2). Note that (3) may be re-written in terms of factorials, since each . A similar result appears in 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.
Wilson's Theorem (which was actually discovered by 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
for 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 if or 2r+1 = p or ,
when the congruence holds ,
where `' can only be `-' if p=2 (which is easily determined
by evaluating both sides of (8) modulo 4), and the integer
except if or 2r+1 = p or , when the congruence holds , where `' can only be `-' if p=2 (which is easily determined by evaluating both sides of (8) modulo 4), and the integer
Note how Theorem 2 allows us to express any in terms of with . In Theorem 3 below we prove a similar result for any factorial, which allows us to compute such factorials very rapidly.
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 then
where is the ith Fermat number.
In section 5 we give somewhat different proofs of these 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 my paper .
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 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 (14) holds modulo if and only if p divides ; however it cannot be modified to investigate other binomial coefficients, and so we use different method for this in section 9. Our first result is that
for any : In fact, this product is whenever the p-2nd Bernoulli polynomial vanishes at (which is immediate for m=2).
In 1938 Emma Lehmer related the values of , for , to Fermat's Last Theorem for exponent p. We shall show, in section 9, that recent, as yet unpublished, results of Skula and Cui-Xiang imply that if the first case of Fermat's Last Theorem is false for prime exponent p (that is, there exist integers a, b, c, not divisible by p, for which ) then
for ; moreover, Skula'a approach implies that the `12' here may be changed to any given number after a finite amount of computation.
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 Hudson and Williams). 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
and this was proved by Chowla, Dwork and Evans. 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 and there are presumably many others waiting to be found.