--- title: Wilson Quotient in Number Theory url: https://www.emergentmind.com/topics/wilson-quotient type: topic --- # Wilson Quotient in Number Theory The Wilson quotient is the integer $$ W_p=\frac{(p-1)!+1}{p}, $$ defined for a prime $p\ge 3$. Its integrality is a direct consequence of Wilson’s theorem, and its arithmetic role is to quantify the extent to which the congruence $(p-1)!\equiv -1\pmod p$ fails to be an exact identity in $\mathbb Z$. In current work, the Wilson quotient appears simultaneously as a classical invariant of primes, a $p$-adic object admitting higher-power refinements, and a node in a wider network of congruences involving left factorials, Bell numbers, Fermat quotients, and generalized Wilson-type quotients [2604.09714] [2509.05235] [2505.16201]. ## 1. Definition and immediate consequences For a prime $p$, Wilson’s theorem states that $$ (p-1)!\equiv -1\pmod p. $$ A standard proof considers the factorization of $x^{p-1}-1$ over $\mathbb Z/p\mathbb Z$: $$ x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p, $$ and evaluates at $x=0$ to obtain $-1\equiv (p-1)!\pmod p$, since $p-1$ is even. An equivalent derivation uses the multiplicative group $(\mathbb Z/p\mathbb Z)^\times$ [2604.09714]. Because $p\mid (p-1)!+1$, the quotient $$ W_p=\frac{(p-1)!+1}{p} $$ lies in $\mathbb Z$. Equivalently, $$ (p-1)!=-1+p\,W_p. $$ This makes $W_p$ the first-order correction term to Wilson’s congruence in the $p$-adic direction. The quantity is often studied modulo powers of $p$, not only modulo $p$ itself, because higher congruence data encode substantially finer information than the classical statement alone [2604.09714] [2510.26743]. A useful characterization is that $$ (p-1)!+1\equiv 0\pmod{p^2} $$ if and only if $p$ is a Wilson prime. This is precisely the condition $W_p\equiv 0\pmod p$ [2604.09714]. ## 2. Wilson primes, elementary examples, and residue-level behavior A prime $p$ is called a Wilson prime if $$ W_p\equiv 0\pmod p. $$ Only three are known: $$ p=5,\;13,\;563. $$ The associated open problem asks whether there are infinitely many Wilson primes or only finitely many; the data under discussion records that this remains open [2604.09714]. For small primes, the quotient is explicit. When $p=3$, $$ W_3=\frac{2!+1}{3}=1. $$ When $p=5$, $$ W_5=\frac{4!+1}{5}=5, $$ so $5$ is a Wilson prime. When $p=7$, $$ W_7=\frac{6!+1}{7}=103. $$ These examples already show that integrality of $W_p$ is universal for primes, whereas vanishing modulo $p$ is exceptional [2604.09714]. The residue class of $W_p$ modulo $p$ is also tied to classical congruences of Lerch type. In particular, later higher-power work recovers $$ W_p\equiv Q_p(1)\pmod p, $$ where $$ q_p(a)=\frac{a^{p-1}-1}{p}, \qquad Q_p(k)=\sum_{a=1}^{p-1} q_p(a)^k. $$ Thus the mod-$p$ Wilson quotient is expressible as a power sum of Fermat quotients, linking Wilson’s theorem to a separate but closely related arithmetic framework [2509.05235]. ## 3. Left factorials, Bell numbers, and the Kurepa–Bell–Wilson congruence The Wilson quotient enters a broader combinatorial-arithmetic structure through the left-factorial sequence and the Bell numbers. For odd primes $p\ge 3$, the left factorial is $$ K_p=\sum_{k=0}^{p-1}k!, $$ while the Bell numbers $\Bell_n$ count set partitions and satisfy the exponential generating function $$ \sum_{n=0}^\infty \Bell_n\frac{x^n}{n!}=\exp\bigl(e^x-1\bigr). $$ Gertsch’s congruence asserts that for every odd prime $p\ge 3$, $$ K_p\equiv \Bell_{p-1}-1\pmod p. $$ According to the cited account, this follows from combining Touchard’s congruences for Bell polynomials with the congruence $(p-1)!\equiv -1\pmod p$ [2604.09714]. Within this setting, the paper "The Wilson Quotient, Left Factorials, Bell Numbers, and the Kurepa–Bell–Wilson Congruence" formulates the Kurepa–Bell–Wilson congruence: $$ \frac{K_p+1}{p}\equiv \frac{\Bell_{p-1}-1}{p}+W_p \pmod p. $$ This congruence makes the Wilson quotient an additive component in a decomposition of a normalized left-factorial expression, with the Bell-number term furnishing the complementary component [2604.09714]. The same work defines the Gertsch quotient $$ \mathbb G_p=\frac{K_p-\Bell_{p-1}+1}{p} =\frac{\sum_{k=0}^{p-1}k!-\Bell_{p-1}+1}{p}\in\mathbb Z. $$ From the KBW congruence one obtains $$ \mathbb G_p\equiv W_p\pmod p. $$ Hence $\mathbb G_p$ is an integer lift of the residue of $W_p$. The paper further interprets this lift in the “poor man’s adele ring” $$ \mathcal A=\Bigl(\prod_p \mathbb Z/p\Bigr)\Big/\Bigl(\bigoplus_p \mathbb Z/p\Bigr), $$ via the natural diagonal embedding of $\mathbb Q$ [2604.09714]. The small-prime instances given in the source illustrate the congruence concretely. For $p=3$, one has $\mathbb G_3=1$ and $W_3=1$. For $p=5$, one has $\mathbb G_5=4$ and $W_5=5$, hence both are congruent modulo $5$. For $p=7$, one has $\mathbb G_7=96$ and $W_7=103$, again with $$ \mathbb G_p\equiv W_p\pmod p. $$ The same source records that no prime $p\ge 3$ is known for which $\mathbb G_p\equiv 0\pmod p$ [2604.09714]. ## 4. Supercongruences and higher prime-power expansions A major contemporary development is the extension of Wilson’s theorem from modulus $p$ to arbitrary higher powers $p^n$. For an odd prime $p$ with $p>n$, Kellner proves that there exist unique multivariate polynomials $$ \psi_\nu(x_1,\dots,x_\nu)\in \mathbb Z[x_1,\dots,x_\nu], \qquad \nu\ge 1, $$ each with no constant term, such that $$ W_p\equiv \sum_{\nu=1}^{n} \frac{p^{\nu-1}}{\nu!}\, \psi_\nu\bigl(Q_p(1),Q_p(2),\dots,Q_p(\nu)\bigr) \pmod{p^n}, $$ and equivalently $$ (p-1)!\equiv -1+ \sum_{\nu=1}^{n} \frac{p^\nu}{\nu!}\, \psi_\nu\bigl(Q_p(1),\dots,Q_p(\nu)\bigr) \pmod{p^{n+1}}. $$ For $n=1$ this reproduces Lerch’s congruence $$ W_p\equiv Q_p(1)\pmod p, \qquad (p-1)!\equiv -1+p\,Q_p(1)\pmod{p^2} $$ [2509.05235]. The construction uses partial Bell polynomials and Newton’s identities. The elementary symmetric sums of the family $\{q_p(1),\dots,q_p(p-1)\}$ are rewritten in terms of the power sums $Q_p(k)$, and the key factorization $$ \prod_{a=1}^{p-1}(1+p\,q_p(a))=(1-p\,W_p)^{p-1} $$ then ties the resulting symmetric-polynomial data to powers of $W_p$. The recursion for the $\psi_\nu$ is uniform in $p$, and the low-degree cases are explicitly $$ \psi_1=x_1,\qquad \psi_2=2x_1-x_1^2-x_2, $$ $$ \psi_3=6x_1-6x_1^2+x_1^3+3x_1x_2-3x_2+2x_3 $$ [2509.05235]. A further refinement computes the Wilson quotient modulo $p^5$ and $p^6$, and equivalently $(p-1)!$ modulo $p^6$ and $p^7$, respectively. In that framework one introduces the divided Bernoulli numbers $$ {}_n=\frac{B_{n(p-1)}+\frac1p-1}{n(p-1)} \quad\text{and}\quad {}_{n,d}=\frac{B_{n(p-1)-d}}{n(p-1)-d} \quad(d=2,4), $$ which lie in $\mathbb Z_p$ and satisfy the usual Kummer congruences. For $p\ge 7$, $$ W_p\equiv \omega_1+\omega_2\,p+\omega_3\,p^2+\omega_4\,p^3+\omega_5\,p^4\pmod{p^5}, $$ and for $p\ge 11$, $$ W_p\equiv \sum_{\nu=1}^6 \omega_\nu p^{\nu-1}\pmod{p^6}. $$ The coefficients $\omega_\nu$ admit closed forms in the divided Bernoulli numbers; for instance, $$ \omega_1\equiv -6\,{}_1+15\,{}_2-20\,{}_3+15\,{}_4-6\,{}_5+{}_6\pmod{p^6} $$ in the modulus-$p^7$ theorem [2510.26743]. The higher-power study also records a regular pattern in the reductions $\omega_\nu\pmod p$: the subsequence of pure powers $-\,{}_1^\nu/\nu!$ appears, along with analogous universal terms in ${}_{1,2}$ and ${}_{1,4}$. The paper formulates the conjectural regularity that for all $\nu$, the reduction $\omega_\nu\pmod p$ is a $\mathbb Q$-linear combination of monomials $$ {}_1^\alpha\,{}_{1,2}^\beta\,{}_{1,4}^\gamma \qquad (\alpha+2\beta+4\gamma=\nu), $$ with rational coefficients depending only on $(\alpha,\beta,\gamma)$ [2510.26743]. ## 5. Generalized Wilson-type quotients The ordinary Wilson quotient is the first member of a larger family arising from an extended corollary of Wilson’s theorem. For any integer $n\ge 1$ and any integer $k$ with $0\le k