1 Introduction
The wellknown Fermat’s Little Theorem states that for any prime p and any integer a,
Wilson’s theorem, on the other hand, states that for any prime p,
Around 1956, Moser [Reference Moser13] discovered the congruence
By setting $a=1$ in (1.3), one obtained (1.2), and using (1.2) in (1.3), we deduced (1.1). On the other hand, (1.3) follows immediately from (1.2) and (1.1). Moser’s congruence (1.3) was mentioned in the book of Sierpinski [Reference Sierpinski20, pp. 216 – 217]. Sierpinski appeared to have mistaken that Moser discovered
when it should have been (1.3).
About 30 years later, Moser’s brother, William Moser [Reference Moser14], found a generalization of Moser’s congruence (1.3) given by
where
When n is a prime p, (1.5) reduces to (1.3). Since the special case of (1.5) is discovered by L. Moser and (1.5) is discovered by his brother, we will name (1.5) the Moser–Moser congruence. The Moser–Moser congruence is a generalization of (1.3), while (1.4) is its analog. In this article, we will derive new generalizations of (1.3) and (1.4).
In 2006, a special case of (1.5) with the value $a=1$ , namely,
was rediscovered by Evans [Reference Evans9] as a generalization of (1.2). It turns out that Evans’ proof of (1.6), which is different from Moser’s proof, can be modified to derive (1.5).
The proofs of Evans and Moser of (1.5) are different. Evans used the action of the group ${\mathbf Z}/n{\mathbf Z}$ on the set of Hamilton cycles with n vertices, whereas Moser used the action of the group ${\mathbf Z}/n{\mathbf Z}\times {\mathbf Z}/n{\mathbf Z}$ on the set of line permutations of $\{1,2,\ldots , n\}$ . In this article, we will present a third proof of (1.6). We will then deduce the Moser–Moser congruence (1.5), which is (1.6) with an extra factor of $a^{n/d}$ , from L. Gegenbauer’s theorem [Reference Gegenbauer10] is given as follows.
Theorem 1.1 Let n and a be positive integers. Let $F:{\mathbf Z}^+\rightarrow {\mathbf Z}.$ Suppose $F(n)\neq \mu (n),$ where $\mu $ is defined in Section 4 (see Definition 4.1). If
then
Although
is true, it does not follow from Theorem 1.1 and the fact [Reference Apostol3, Theorem 2.1] that
for $n>1$ . This is because (1.7) is used in the proof of Theorem 1.1 and (1.7) needs to be proved separately as what Gegenbauer did in his article.
In the next section, we state the Frobenius–Burnside theorem, a fundamental tool used in proving many congruences such as (1.5) and (1.7). In Section 3, we give a new proof of the Moser–Moser congruence (1.5) using the Frobenius–Burnside. In Section 4, we give a new proof of an interesting result of András [Reference András1, p. 267, Remark 2] (see Theorem 4.3) and give a brief history of (1.7) and its analogue. In Section 5, we give a generalization of (1.5). In the final section, we prove a generalization of Theorem 1.1.
In this article, several congruences such as (1.5) are stated modulo $n^2$ , whereas Theorems such as Theorem 1.1 hold modulo n. We emphasize that in all our examples, we derive
using theorems such as Theorem 1.1 and conclude that
We now summarize our contributions. In this article, we present a new proof of the Moser–Moser congruence with the help of Theorem 1.1. Surprisingly, this theorem has remained relatively obscure despite its significance. We have also found the origin of (1.6). We have identified an error in Sierpinski’s book, where he mistakenly attributed (1.4) instead of (1.3) to L. Moser. Additionally, we have unveiled a comprehensive generalization that encompasses both (1.3) and (1.4) (see Section 5). By using Theorem 4.3 and Theorem 6.1, we derived many new congruences from wellestablished ones, such as Moreau’s congruence.
We would like to thank the anonymous referee for his/her invaluable insights and the references [Reference Byszewski, Graff and Ward4, Reference Wójcik23]. These references establish crucial connections between our work and Dold sequences. Notably, one can see a link between our generalization of the Gegenbauer theorem (Theorem 6.1) and [Reference Wójcik23, Theorem 7]. This connection becomes apparent when we recognize that the function $G(n)$ defined in Theorem 6.1 is Dold sequences, leading to the conclusion that Theorem 8 follows from the equivalence of the statements in [Reference Wójcik23, Theorem 7]. This connection also shows that many of the examples discussed in this article are, therefore, examples of Dold sequences. Some of the sequences contained in (6.4) and in (6.6) might pose challenges for direct verification as Dold sequences under alternative circumstances. Finally, as highlighted by the referee, several previous works, including [Reference András1, Reference Wójcik23], have overlooked the significance of the Gegenbauer theorem (Theorem 1.1). Our article serves to rectify this gap in the mathematical discourse.
2 The Frobenius–Burnside theorem
Let $(G,\circ )$ be a group with identity $1_G$ , and let X be a nonempty set. A left action of the group G on X is a function
which we write $g\bullet x$ instead of $\bullet (g,x)$ , with the following properties that for all $a,b\in G$ and $x\in X$ :

(1) $a\bullet (b\bullet x) = (a\circ b)\bullet x,$

(2) $1_G\bullet x = x.$
We say that G acts on X via $\bullet $ .
Given a group G acting (via $\bullet $ ) on a nonempty set X, we define a relation $\sim $ as follows:
The relation $\sim $ is an equivalence relation on X. This implies that X is a disjoint union of equivalence classes $\mathcal O_x$ , which are also called Gorbits, where
When X is a finite set, the number of distinct Gorbits in X is given by the following theorem.
Theorem 2.1 (The Frobenius–Burnside theorem)
Let G be a group acting on a finite set X, and let
The number of disjoint Gorbits N in X is given by
where $S$ denotes the number of elements in the set S.
We emphasize that in the proof of Theorem 2.1, one has to show that if
then
This shows immediately that
For a complete proof of the Frobenius–Burnside theorem, see [Reference Robinson17, Chapter 5].
3 A new proof of the Moser–Moser congruence
For any positive integer n, let $S_n$ denote the set of bijections on
For $g \in S_n$ , let
be the complete factorization (which is unique up to the arrangement of the cycles) of g into disjoint cycles. If $\tau _j$ is a cycle with length $\lambda _j$ , then we say that g has cycle type
If g has $a_j$ cycles of length j, we say that g has cycle structure
For example, if
then the cycle type and cycle structure of g are $(3, 2, 2, 2)$ and $\langle 1^0 2^{3}3^{1}\rangle $ , respectively.
The following lemma is well known [Reference Cameron5, p. 244].
Lemma 3.1 Let $g\in S_n$ with cycle structure $\langle 1^{a_1}2^{a_2} \cdots n^{a_n}\rangle $ , and let
be the centralizer of g in $S_n$ . Then
We will use Lemma 3.1 to derive the following lemma.
Lemma 3.2 Suppose $g, h\in S_n$ . Let $F(g,h) = \{x \in S_ng^{1}xh = x\}$ . Then $F(g,h) $ is nonempty if and only if g and h have the same cycle type, or equivalently, g and h are conjugate to each other in $S_n$ . Moreover, if $F(g,h)$ is nonempty, then
where $\langle 1^{a_1}2^{a_2} \cdots n^{a_n}\rangle $ is the cycle structure of g (and h).
Proof If g and h are conjugates, then $h=y^{1}gy$ for some $y\in S_n$ and therefore $y\in F(g,h)$ and $F(g,h)$ is nonempty. If $F(g,h)$ is nonempty, then there exists $y\in F(g,h)$ such that $g^{1}yh=y$ or $g=y^{1}hy$ , and so g and h are conjugates.
Suppose $h = y^{1}g y$ for some $y \in S_n$ . We claim that if $z\in C(g)$ , then $zy \in F(g,h)$ . We need only to check that
since $g^{1}z = zg^{1}$ . Define a map
by $\theta (z) = zy,$ which is valid since $zy\in F(g,h)$ . Clearly, $\theta $ is injective since $zy = z'y$ implies that $z=z'$ .
Next, for $x \in F(g,h)$ , note that $g^{1}xh=x,$ or $xhx^{1}=g$ . Let $z=xy^{1}$ . Then
and therefore $z\in C(g)$ . It is clear that $\theta (z)=\theta (xy^{1})=xy^{1}y=x$ and so $\theta $ is surjective. The second assertion now follows from Lemma 3.1.
We are now ready to give a new proof of (1.6).
Proof of (1.6)
Let $C_n$ be the subgroup of $S_n$ generated by the ncycle $(1\ 2\ \cdots \ n).$ Consider the action $G=C_n \times C_n$ on $X = S_n$ given by $(g, h) \bullet x = g^{1}x h$ . Indeed, this is a group action since
By Theorem 2.1, the number of orbits N arising from G acting on $S_n$ is
Note that $\text {Fix}((g,h))=F(g,h)$ where $F(g,h)$ is the set defined in Lemma 3.2.
For an element $g\in G$ , we will use $o(g)$ to denote the order of g in G. We have seen that $F(g,h)$ is nonempty if and only if g and h are conjugate to each other. In other words, $F(g,h)$ is nonempty if and only if $o(g)=o(h)$ . Furthermore, $o(g)n$ . Therefore, we may rewrite (3.1) as
Next, it is known that if $o(g)=d$ in $C_n$ , then g has cycle type $c(g)=( d,d,\ldots , d),$ where there are $n/d$ copies of d in $c(g)$ . By Lemma 3.2, we conclude that
and rewrite (3.2) as
since there are precisely $\varphi ^2(d)$ elements $(g,h)\in G$ with $o(g)=o(h).$ The congruence (1.6) now follows immediately.
This new proof of (1.6) explains the presence of $(n/d)! d^{n/d}$ and $\varphi ^2(d)$ in (1.6).
This section is motivated by the articles of Evans [Reference Evans9] and András [Reference András1]. We are led to these articles because of our interest in generalizing Petersen’s proof of Wilson’s theorem [Reference Petersen15] found in Andrews’ book [Reference Andrews2, Section 3.3].
4 Some facts about arithmetic functions and András’ theorem
An arithmetic function f is a function from the set of positive integers to the set of complex numbers. One of the most important arithmetic functions is the Möbius function defined as follows.
Definition 4.1 Let $\mu (1)=1$ and for $\displaystyle n=\prod _{k=1}^m p_k^{\alpha _k},$ let
The Dirichlet product of two arithmetic functions is a binary operation defined as follows.
Definition 4.2 Let f and g be two arithmetical functions. We define the Dirichlet product of f and g, denoted by $f*g$ , as
From now on, we will use $f*g(n)$ to represent $(f*g)(n)$ , removing the use of brackets. We will record a few facts regarding several arithmetic functions that we need for this article.
Lemma 4.1 For any $n\in {\mathbf Z}^+$ , let $u(n)=1$ , $\mathcal N(n)=n$ , and $I(n)=[1/n],$ where $[\cdot ]$ is the greatest integer function. Then:

(1) $u*\mu (n)=I(n),$

(2) $\mu *{\mathcal N}(n) =\varphi (n),$

(3) $\mu {\mathcal N} * {\mathcal N}(n)=I(n).$
The proofs of Lemma 4.1(a)–(c) can be found in Apostol’s book [Reference Apostol3, Theorem 2.1], [Reference Apostol3, Theorem 2.3], and [Reference Apostol3, Theorem 2.17], respectively.
Lemma 4.2 Let f and g be two arithmetic functions. Then
if and only if
The proof of Lemma 4.2 can be found in [Reference Apostol3, Theorem 2.9].
Around 2011, András [Reference András1] revisited Evans’ work and discovered the following interesting result.
Theorem 4.3 Let $F(n)$ be an arithmetic function. Then
if and only if
András has a complete proof of the above, but (4.1) implies (4.2) was not presented in his article. We now give our proof of Theorem 4.3 using Lemma 4.1.
Proof Suppose (4.1) holds. Then there exists an integervalued arithmetic function G such that
This implies that
Applying Lemma 4.1(b) followed by Lemma 4.1(c), we may rewrite the lefthand side of (4.3) as
The righthand side of (4.3) is
Therefore, (4.2) holds.
Next, suppose (4.2) holds. Then
for some integervalued arithmetic function G. This implies that
Using Lemma 4.1(b), we deduce that the lefthand side of (4.4) is
whereas the righthand side of (4.4) is
Therefore, (4.1) holds.
András used Theorem 4.3 to deduce from (1.6) that
We observe that András’ congruence is also a generalization of (1.3). If we apply Theorem 4.3 to (1.5), we deduce the new congruence
Applying Theorem 4.3 one more time to (4.5), we conclude that
Both congruences (4.5) and (4.6) are also generalizations of (1.3).
In the next section, we give a generalization of (4.5) and hence generalizations of (1.5) and (4.6) using Theorem 4.3.
With Theorem 4.3, we can now conclude that the congruences
and (1.7) are equivalent. Both (4.7) and (1.7) are generalizations of (1.1). We will now give a brief history of (4.7) and (1.7). Congruence (4.7) was discovered by MacMahon [Reference MacMahon11] using the following result of Moreau [Reference Moreau12, p. 313].
Theorem 4.4 Let n be a positive integer, and let
where $\alpha _1,\alpha _2,\ldots ,\alpha _s$ are positive integers. Let $D=\text {gcd}(\alpha _1,\alpha _2,\ldots , \alpha _s).$ Then
The works of MacMahon and Moreau are mentioned in Riordan’s book [Reference Riordan16, p. 162, Problem 37]. The approach to these congruences in [Reference Riordan16] involves using the Redfield–Polya theorem (see Stanley’s book [Reference Stanley21, p. 394, Theorem 7.24.4]). Another proof of (4.7) was given by Gegenbauer [Reference Gegenbauer10]. He used Theorem 1.1 and the identity [Reference Apostol3, p. 26, Theorem 2.2]
to derive (4.7).
We now discuss (1.7). Congruence (1.7) in the case when a is a prime number is probably known to C.F. Gauss. The expression
counts the number of irreducible polynomials of degree n over the finite field of p elements (see Cox’s book [Reference Cox6, p. 300, Theorem 11.2.4]). This means that (1.7), after applying Dirichlet’s theorem of primes in arithmetic progression, is true for any integer n such that $(a,n)=1$ . According to Dickson [Reference Dickson7, p. 84], (1.7) was first stated by Serret [Reference Serret19, p. 262] in 1855. One of the simplest proofs of (1.7) can be found in Szele’s article [Reference Szele22] and Gegenbauer’s article [Reference Gegenbauer10]. See also an exercise in Rose’s book [Reference Rose18, p. 55, Problem 11].
Another proof of (1.7), using Theorem 2.1, relies on counting the number of orbits with exactly n elements for the group ${\mathbf Z}/n{\mathbf Z}$ acting on X, the set of all necklaces obtained from beads which can be colored using a colors. Note that $X=a^n$ . Let $N_m$ be the number of orbits with exactly m elements. Note that by (2.1), $N_m$ is nonempty if and only if $mn$ . Therefore,
Applying Lemma 4.2, we deduce that
and (1.7) follows. For yet another proof of (1.7) using periodic points of a function, see [Reference András1, p. 268].
5 A generalization of the Moser–Moser congruence
In this section, we derive the following generalization of the Moser–Moser congruence.
Theorem 5.1 Let $n, a$ , and b be positive integers. Then
It is immediate that (1.5) is a special case of (5.1) with $b=1$ .
By Theorem 1.1, we see that congruence (5.1) follows immediately from the following.
Theorem 5.2 Let n and b be positive integers. Then
We should note the difference between (5.2) and (1.5). Our congruence (5.2) is a generalization of (1.4) in exactly the same manner that (1.5) is a generalization of (1.3). We have not been able to find a proof of (5.2) which involves Theorem 2.1.
Proof Let $\displaystyle n=\prod _{j=1}^s p_j^{\alpha _j}.$ Our aim is to show that for $1\leq j\leq s$ ,
We note that it suffices to prove (5.3) for a fixed j. Let $p=p_1$ and $\alpha =\alpha _1$ . We first observe that the presence of $\mu (n)$ implies that
We pair up the divisors of $pp_2\cdots p_s$ as $p\nu $ and $\nu $ , where $(\nu ,p)=1$ . We claim that for each such pair of divisors, the terms corresponding to them satisfy
where $\ell $ is given by $\ell =n/p^{\alpha }$ . Congruence (5.4) would imply (5.3). Note that from (5.4), we find that
Let
It suffices to show that
We divide our arguments into the following cases, according to the values of $\ell /\nu $ , $\alpha $ , and p.
Case 1: $\ell /\nu \geq 2$ . We note that $(p^{\alpha } \ell /\nu )! $ contains the factors $2p^{\alpha }$ and $p^{\alpha }$ . Therefore, $p^{2\alpha }$ is a factor of $S_1$ . On the other hand, we see that
Therefore, $p^{2\alpha }$ divides $p^{p^{\alpha 1} \ell /\nu }$ , and so is a factor of $S_2$ . Hence, (5.5) is true.
Case 2a: $\ell /\nu =1$ and $\alpha =1$ . In this subcase, we find that
where we have used Wilson’s theorem and Fermat’s Little Theorem.
Case 2b(i): $\ell /\nu =1$ , $\alpha =2$ and $p=2$ . For these values, we see that
since $\nu ^2a^{\nu }(3\nu ^2a^{\nu })$ must be even.
Case 2b(ii): $\ell /\nu =1$ , $\alpha =2$ , and $p\geq 3$ . We note that $(p^2)!$ contains the factors $p^2$ , $2p$ , and p. Hence, $p^4$ divides $S_1$ . On the other hand, clearly, $p^3$ divides $(p\nu )^p$ and so $p^4$ divides $S_2$ .
Case 2c: $\ell /\nu =1$ and $\alpha \geq 3$ . We note that $(p^{\alpha })!$ contains the factors $p^{\alpha }$ , $p^{\alpha 1}$ , and p. Hence, $p^{2\alpha }$ divides $S_1$ . On the other hand,
Therefore, $p^{2\alpha 2}$ divides $p^{p^{\alpha 1}}$ . Also, clearly, $p^2p^{\alpha 1}$ . Collectively, we see that $p^{2\alpha }$ divides $S_2$ .
Thus, all cases of the congruence (5.5) are proved, and this completes the proof of the theorem.
Once again, by Theorem 4.3, we deduce the following theorem.
Theorem 5.3 Let n and b be positive integers. Then
By using Theorem 4.3 yet again, we deduce from (5.6) the following congruence:
6 Analogues of Gegenbauer’s theorem
In this final section, we will derive a generalization of Gegenbauer’s theorem.
Theorem 6.1 Let n be a positive integer. Let $F:{{\mathbf Z}}^+\rightarrow {\mathbf Z}$ and G be an arithmetic function. Suppose
If
then
The function $G(n)$ which appears in Theorem 6.1 is known as a Dold sequence [Reference Byszewski, Graff and Ward4, Definition 2.1].
Let a be a positive integer, and let
Then since (1.7) holds, we see that Theorem 6.1 implies Theorem 1.1.
Our proof of Theorem 6.1 follows Gegenbauer’s proof of Theorem 1.1. Another proof of Theorem 6.1 can be found in a recent article of Wójcik [Reference Wójcik23]. The proof given there is dependent on the equivalence of Newton sequence and Dold sequence established by Du, Huang, and Li [Reference Du, Huang and Li8].
Proof of Theorem 6.1
and
respectively, where $H(\ell ), K(\ell )\in {\mathbf Z}$ for all $\ell \in {\mathbf Z}^+$ . From (6.3) and Lemma 4.2, we deduce that
This implies that
and the proof of Theorem 6.1 is complete.
There are already several $G(n)$ which we can choose from the congruences we discussed in the previous sections. For example, we could identify $G(n)$ from (5.6) and deduce using Theorem 6.1 that if (6.2) holds, then
We may also apply Theorem 6.1 to a variant of Theorem 4.4, which we state as follows (with $\varphi $ replaced by $\mu $ via Theorem 4.3).
Theorem 6.2 Let n be a positive integer, and let
where $\alpha _1,\alpha _2,\ldots ,\alpha _s$ are positive integers. Let $D=\text {gcd}(\alpha _1,\alpha _2,\ldots , \alpha _s).$ Then
The function $G(n)$ in (6.1) can now be identified from (6.5), allowing us to deduce from Theorem 6.1 that if (6.2) holds, then
We may use combinations of Theorem 1.1, (6.4), and (6.6) to derive many new congruences. For example, we have
Acknowledgment
We are very grateful to Professors G. E. Andrews, W. Zudilin, and KuoJye Chen for their encouragement. We thank Professor P. Moree for helping us locate L. Moser’s paper. We also thank Professors T. J. Evans and S. András for answering our queries with regard to their works. Finally, we thank the referee for the references [Reference Byszewski, Graff and Ward4, Reference Wójcik23].