I am writing an article on Fermat's work in number theory and feel uncomfortable everytime I have to write "Fermat's Little Theorem": it's clumsy and belittles the fundamental character of Fermat's result. "Fermat's Theorem" is too ambiguous, and I don't really like acronyms such as Flt or Flit. Has anyone ever seen a better name for this result (or a new suggestion)?
[Math] Christening Fermat’s Little Theorem
ho.history-overviewnt.number-theory
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Here's a massive generalization, with references.
Consider a sequence $\{a_n \}_{n\ge 1}$ of integers. We'll say it satisfies the necklace congruences if $a_{n} \equiv a_{n/p} \mod n$ whenever $p \mid n$ ($p$ prime).
Here are some equivalent formulations:
- $a_{p^{k+1} m} \equiv a_{p^k m} \mod {p^{k+1}}$ for all $p,m,k$
- $\sum_{d \mid n} a_d \mu(n/d) \equiv 0 \mod n$ for all $n$
- The Mobius function can be replaced by any arithmetic function $f$ satisfying $\sum_{d \mid n} f(d) =0 \mod n, f(1) = \pm 1$.
So any such sequence gives rise to an integer sequence $\lambda_{\tilde{a}}(n) := \frac{1}{n}\sum_{d \mid n} a_d \mu(n/d)$ which satisfies $a_n = \sum_{d \mid n} d \lambda_{\tilde{a}}(n)$ by Mobius inversion. One can then define $\sigma_{\tilde{a}}(n):=\sum_{d \mid n, 1<d<n} d\lambda_{\tilde{a}}(d)$ and obtain the following generalization of Fermat's little theorem:
$a_n \equiv a_1 + \sigma_{\tilde{a}}(n) \mod n$
Now, let's go back to necklace congruences. A less obvious equivalence is the following:
Theorem: $\{a_n\}_{n \ge 1}$ satisfies the necklace congruences iff $\zeta_{a}(x):=\exp(\sum_{n\ge1} \frac{a_n}{n}x^n)$ has integer coefficients.
Proof: Write, $\zeta_a$ formally as $\prod_{n\ge 1} (1-x^n)^{-\frac{b_n}{n}}$. The $b_n$'s are uniquely determined, and in fact (by taking logarithmic derivative) it can be seen that they are $b_n = \sum_{d \mid n} a_d \mu(n/d)$. Now one direction is immediate and the other requires an inductive argument. $\blacksquare$
(Reference: Exercise 5.2 (and its solution) in Richard Stanley's book "Enumerative Combinatorics, vol. 2")
As Sergei remarked, for any integer square matrix $A$, the sequence $a_n := \text{Tr}(A^n)$ satisfies the necklace congruences. This is immediate from the last theorem, as the corresponding "zeta" function $\zeta_a$ is just:
$\exp( \sum_{n \ge 1} \frac{\text{Tr}((Ax)^n)}{n}) = \exp (\text{Tr} (- \ln (I -Ax)))=\det(I-Ax)^{-1} \in \mathbb{Z}[x]$
In particular, by taking a 1 on 1 matrix, we recover your observation.
Another generalization is $a_n = [x^n]f^n(x)$, where $f \in \mathbb{Z}[[x]]$. By taking the linear polynomial $f(x)=1+ax$ we recover your example, by taking $f(x)=(1+x)^m$ we get $a_n = \binom{nm}{n}$.
The really interesting feature is that this notion has a local version. Specifically, theorems of Dieudonne-Dwork and Hazewinkel give the following result:
Theorem: Given $\{ a_n \}_{n\ge 1} \subseteq \mathbb{Q}_{p}$ ($p$-adic numbers), the following are equivalent:
- $\zeta_{a}(x) := \exp(\sum_{n\ge 1} \frac{a_n }{n}x^n)\in\mathbb{Z}_{p}[x]$
- $\exp( \sum_{n \ge 1} p \frac{a_n - a_{n/p}}{n} x^n) \in 1+px\mathbb{Z}_{p}[x]$ (I defnite $a_{n/p}=0$ if $p\nmid n$)
- $\sum_{n \ge 1} \frac{a_n - a_{n/p}}{n} x^n) \in \mathbb{Z}_{p}[x]$
- $p \mid n \implies a_n \equiv a_{n/p} \mod {n\mathbb{Z}_{p}}$
Applying this simultaneously for all $p$, we recover the previous theorem. A reference for this theorem is "A Course in p-adic Analysis" by Alain M. Robert (the theorems are stated there in a much more general context).
A proof is essentially given in Section 5.1 of Notes on primitive lambda-roots by P. J. Cameron and D. A. Preece.
Best Answer
I think you shouldn't change the name. It's universally known as Fermat's Little Theorem, and especially if you're writing a survey or historical article, you're not in a place to try to revolutionize established mathematical nomenclature. There are many instances of unfortunate terminology in mathematics, but in my opinion, once they are in general use, they become part of the lore and the culture. I would make exceptions only in a few cases, such as:
a) it's on the level of adjectives such as "good" and "admissible", b) it's crediting the wrong person (Cayley numbers, Burnside's lemma), or c) it's very recent, with the inventor implicitly begging to attach his name to it
And if your life work is going to become known as "Lemmermeyer's dirty trick", well, take it with humor.