Tag Archives: Generating functions

Generalizations of Fermat’s Little Theorem and combinatorial zeta functions

Everyone who studies elementary number theory learns two different versions of Fermat’s Little Theorem:

Fermat’s Little Theorem, Version 1: If p is prime and a is an integer not divisible by p, then a^{p-1} \equiv 1 \pmod{p}.

Fermat’s Little Theorem, Version 2: If p is prime and a is any integer, then a^{p} \equiv a \pmod{p}.

as well as the following extension of Version 1 of Fermat’s Little Theorem to arbitrary positive integers n:

Euler’s Theorem: If n is a positive integer and (a,n)=1, then a^{\phi(n)} \equiv 1 \pmod{n}, where \phi is Euler’s totient function.

My first goal in this post is to explain a generalization of Version 2 of Fermat’s Little Theorem to arbitrary n. I’ll then explain an extension of this result to m \times m integer matrices, along with a slick proof of all of these results (and more) via “combinatorial zeta functions”.

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Infinitely many primes via generating functions

A few years ago I discovered an amusing proof of Euclid’s theorem that there are infinitely many primes which I thought I’d record here for posterity. (I subsequently learned that a similar argument appears in this paper by Paul Pollack.)

To motivate the proof, and illustrate its working in an (admittedly silly) special case, suppose that there were just two prime numbers, called p and q. Then by the fundamental theorem of arithmetic (i.e., unique factorization into primes) we would have the following identity between generating functions:

\sum_{n=2}^\infty z^n = \sum_{k=1}^\infty z^{kp} + \sum_{k=1}^\infty z^{kq} - \sum_{k=1}^\infty z^{kpq}.

Indeed, there is precisely one term z^n for each integer n \geq 2 on the left-hand side, and the same is true for the right-hand side (consider separately the cases where p \mid n but q \nmid n, q \mid n but p \nmid n, and pq \mid n). Using the formula for the sum of a geometric series, we can rewrite our formula as an identity between complex-analytic functions valid on the open unit disc {\mathbb D} = \{z \in {\mathbb C} \; : \; |z|<1 \}:

\frac{z^2}{1-z} = \frac{z^p}{1-z^p} + \frac{z^q}{1-z^q} - \frac{z^{pq}}{1-z^{pq}}.

This is impossible, however, as we see by letting z approach a primitive pq^{\rm th} root of unity, since each term stays bounded except for \frac{z^{pq}}{1-z^{pq}}, which tends to infinity.

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The Drunkard’s Walk and Catalan Numbers

In this post, I’d like to present an amusing and off-the-beaten-path solution to the classical “Drunkard’s Walk” problem which at the same time derives the well-known generating function for the Catalan numbers.  This solution evolved from a suggestion by my former undergraduate student Stefan Froehlich during a discussion in my Math 4802 (Mathematical Problem Solving) class at Georgia Tech in Fall 2007.

In case you’re not familiar with it, here’s the problem (in a formulation I learned from this book by Frederick Mosteller):

A drunken man standing one step from the edge of a cliff takes a sequence of random steps either away from or toward the cliff. At any step, his probability of taking a step away from the cliff is p (and therefore his probability of taking a step toward the cliff is 1-p). What is the probability that the man will eventually fall off the cliff?

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