--- title: 'Wilson''s Theorem: Prime Criterion' url: https://www.emergentmind.com/topics/wilson-s-theorem type: topic --- # Wilson's Theorem: Prime Criterion Wilson’s theorem is the congruence \[ (p-1)!\equiv -1 \pmod p \] for prime \(p\), equivalently the primality criterion \[ n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr) \] for integers \(n\ge2\). It identifies primality through a factorial residue and has become a prototype for congruential primality criteria, \(p\)-adic refinements, Stirling-number generalizations, and finite-field product formulas [1911.09492]. The congruence was discovered, in somewhat different language, by Lagrange in 1771, and later appeared under Wilson’s name [1809.04767]. ## 1. Classical formulation and arithmetic content In its standard form, Wilson’s theorem asserts that if \(p\) is prime, then \((p-1)!\equiv -1\pmod p\), and the converse also holds: if \((n-1)!\equiv -1\pmod n\), then \(n\) is prime [1809.04767]. The theorem therefore gives an exact congruential characterization of primality. A common sharpening separates the prime and composite cases more explicitly. For integers \(n\ge2\), \[ (n-1)!\equiv \begin{cases} -1\pmod n,& n\text{ prime},\\ 0\pmod n,& n\text{ composite},\ n\neq4, \end{cases} \] with the unique composite exception \(n=4\), since \(3!=6\equiv2\pmod4\) [1302.3676]. This formulation isolates the exceptional behavior of the modulus \(4\) and clarifies that the factorial is usually annihilated modulo a composite modulus. The converse direction is elementary and structurally important. If \(n>1\) is composite and \(d\) is a nontrivial divisor of \(n\), then \(d\mid (n-1)!\), so \((n-1)!+1\not\equiv0\pmod d\), hence \((n-1)!\not\equiv-1\pmod n\) [1911.09492]. The prime case is therefore the genuinely nontrivial part of the theorem. ## 2. Proof architectures The most classical proof uses the multiplicative group \((\mathbb Z/p\mathbb Z)^\times\). Every residue \(a\not\equiv a^{-1}\pmod p\) pairs with its inverse, and each pair contributes \(1\) to the total product. The only unpaired elements are those satisfying \(a^2\equiv1\pmod p\), namely \(a=1\) and \(a=p-1\equiv-1\), so \[ (p-1)!\equiv 1\cdot(-1)\equiv -1\pmod p \] [1911.09492]. This proof makes explicit that the theorem is a statement about the involutive structure of the unit group. A second proof is polynomial. Over \((\mathbb Z/p\mathbb Z)[X]\), Fermat’s little theorem implies that every nonzero residue is a root of \(X^{p-1}-1\), hence \[ X^{p-1}-1=\prod_{k=1}^{p-1}(X-k). \] Comparing constant terms yields \[ -1\equiv (-1)^{p-1}(p-1)!\equiv (p-1)!\pmod p \] [1912.06652]. A closely related formulation is \[ \prod_{a=1}^{p-1}(x-a)\equiv x^{p-1}-1 \pmod p, \] which is the starting point for several later higher-power refinements [2509.05235]. A third proof, closer to Lagrange’s original style, expands \[ (x-1)(x-2)\cdots(x-(n-1)) =\sum_{k=0}^{n-1}(-1)^kP_kx^{\,n-1-k}, \] where \(P_k\) is the \(k\)th elementary symmetric sum in \(1,\dots,n-1\). A coefficient comparison after shifting the variable produces the recurrence \[ (m-1)\,P_{m-1}=\sum_{k=0}^{m-2}P_k\binom{n-k}{m-k}, \] and when \(n=p\) is prime this recurrence yields \(p\mid P_1,\dots,P_{p-2}\). Setting \(m=p\) then gives \((p-1)!\equiv-1\pmod p\) [1809.04767]. This approach is notable because the same symmetric-sum machinery also leads to Wolstenholme’s theorem. ## 3. Wilson quotient and higher prime powers Wilson’s theorem can be reframed through the Wilson quotient \[ W_p=\frac{(p-1)!+1}{p}, \] which is integral exactly because \((p-1)!\equiv-1\pmod p\) [2509.05235]. This reformulation converts the basic congruence into a \(p\)-adic object and opens the way to supercongruences. A central auxiliary notion is the Fermat quotient \[ q_p(a)=\frac{a^{p-1}-1}{p}, \] defined for \(\gcd(a,p)=1\), together with its power sums \[ Q_p(n)=\sum_{a=1}^{p-1} q_p(a)^n. \] Lerch’s congruence states that \[ W_p\equiv Q_p(1)\pmod p \] [2509.05235]. Kellner’s later work lifts this to arbitrary higher prime powers by combining symmetric-function identities, Bell polynomials, and Newton’s identities. For every \(n\ge1\) and every odd prime \(p>n\), there are unique integer polynomials \(\psi_1,\dots,\psi_n\) with no constant term such that \[ W_p\equiv \sum_{\nu=1}^{n}\frac{p^{\nu-1}}{\nu!}\, \psi_\nu\bigl(Q_p(1),\ldots,Q_p(\nu)\bigr)\pmod{p^n}, \] and equivalently \[ (p-1)!\equiv -1+\sum_{\nu=1}^{n}\frac{p^\nu}{\nu!}\, \psi_\nu\bigl(Q_p(1),\ldots,Q_p(\nu)\bigr)\pmod{p^{n+1}} \] [2509.05235]. An earlier higher-power milestone is Glaisher’s congruence, recovered by Levaillant through \(p\)-adic lifting and Faulhaber polynomials: \[ (p-1)!\equiv -1+p\,B_{p-1}\pmod{p^2}. \] The same method also yields a formula for \((p-1)!\) modulo \(p^3\) in terms of Bernoulli numbers [1912.06652]. Kellner’s second and third papers translate the \(\psi_\nu\)-expansions into explicit congruences involving divided Bernoulli numbers and Kummer congruences, obtaining \(W_p\) modulo \(p^4\) and equivalently \((p-1)!\) modulo \(p^5\), then extending further to \(W_p\) modulo \(p^5\) and \(p^6\), and equivalently \((p-1)!\) modulo \(p^6\) and \(p^7\) [2509.05402; 2510.26743]. | Paper | Explicit reach | Main mechanism | |---|---|---| | [1912.06652] | \((p-1)!\) mod \(p^2\), \(p^3\) | Hensel lifting, Faulhaber polynomials, Bernoulli numbers | | [2509.05402] | \(W_p\) mod \(p^4\), \((p-1)!\) mod \(p^5\) | \(\psi_\nu\)-polynomials, Bernoulli polynomials, Kummer congruences | | [2510.26743] | \(W_p\) mod \(p^5\), \(p^6\); \((p-1)!\) mod \(p^6\), \(p^7\) | Fermat-quotient power sums, divided Bernoulli numbers | These developments show that classical Wilson is the first layer of a much richer \(p\)-adic expansion, rather than an isolated mod-\(p\) coincidence. ## 4. Stirling-theoretic and combinatorial generalizations A substantial generalization arises from Stirling numbers of the second kind. For positive integers \(m,n\) and complex \(z\), the Stirling function \[ S(m,n,z):=(-1)^d\,n!\sum_{k=0}^n \binom{n}{k}(-1)^k(z-k)^m, \qquad d=m-n, \] satisfies \[ S(m,n,z)-S(m,n)=(-1)^dP_{(m,n)}(z) \] when \(d>0\), where \[ P_{(m,n)}(z)=\sum_{j=1}^d \binom{m}{j}S(m-j,n)(-z)^j. \] With \(B(m,n):=n!\,S(m,n)\), the main arithmetic result is: \[ p\text{ is prime}\quad\Longleftrightarrow\quad -1\equiv_p B(n(p-1),p-1)\ \text{for every }n\in\mathbb Z_+ \] [1701.00044]. The case \(n=1\) gives \[ B(p-1,p-1)=(p-1)!, \] so classical Wilson is recovered exactly. The same framework yields complementary congruences for Stirling numbers. If \(p>2\) is prime, then for every \(n\in\mathbb Z_+\) and every \(0