Wilson's Theorem: Prime Criterion

• Wilson's Theorem is a fundamental result in number theory that defines prime numbers via the congruence (p-1)! ≡ -1 mod p.
• It is proven using group-theoretic and polynomial methods which clearly illustrate the special role of multiplicative inverses in modular arithmetic.
• Extensions include Wilson quotients and generalizations involving Stirling numbers, Bernoulli numbers, and factorial variants in finite fields.

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 (Sauvaget, 2019). The congruence was discovered, in somewhat different language, by Lagrange in 1771, and later appeared under Wilson’s name (Hussein, 2018).

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 (Hussein, 2018). The theorem therefore gives an exact congruential characterization of primality.

A common sharpening separates the prime and composite cases more explicitly. For integers $n\ge2$,

$p$0

with the unique composite exception $p$1, since $p$2 (Aebi et al., 2013). This formulation isolates the exceptional behavior of the modulus $p$3 and clarifies that the factorial is usually annihilated modulo a composite modulus.

The converse direction is elementary and structurally important. If $p$4 is composite and $p$5 is a nontrivial divisor of $p$6, then $p$7, so $p$8, hence $p$9 (Sauvaget, 2019). The prime case is therefore the genuinely nontrivial part of the theorem.

2. Proof architectures

The most classical proof uses the multiplicative group $$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$0. Every residue $$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$1 pairs with its inverse, and each pair contributes $$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$2 to the total product. The only unpaired elements are those satisfying $$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$3, namely $$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$4 and $$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$5, so

$$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$6

(Sauvaget, 2019). This proof makes explicit that the theorem is a statement about the involutive structure of the unit group.

A second proof is polynomial. Over $$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$7, Fermat’s little theorem implies that every nonzero residue is a root of $$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$8, hence

$$n\in\mathcal P \quad\Longleftrightarrow\quad n\mid \bigl(1+(n-1)!\bigr)$$9

Comparing constant terms yields

$n\ge2$0

(Levaillant, 2019). A closely related formulation is

$n\ge2$1

which is the starting point for several later higher-power refinements (Kellner, 5 Sep 2025).

A third proof, closer to Lagrange’s original style, expands

$n\ge2$2

where $n\ge2$3 is the $n\ge2$4th elementary symmetric sum in $n\ge2$5. A coefficient comparison after shifting the variable produces the recurrence

$n\ge2$6

and when $n\ge2$7 is prime this recurrence yields $n\ge2$8. Setting $n\ge2$9 then gives $p$0 (Hussein, 2018). 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

$p$1

which is integral exactly because $p$2 (Kellner, 5 Sep 2025). This reformulation converts the basic congruence into a $p$3-adic object and opens the way to supercongruences.

A central auxiliary notion is the Fermat quotient

$p$4

defined for $p$5, together with its power sums

$p$6

Lerch’s congruence states that

$p$7

(Kellner, 5 Sep 2025). Kellner’s later work lifts this to arbitrary higher prime powers by combining symmetric-function identities, Bell polynomials, and Newton’s identities. For every $p$8 and every odd prime $p$9, there are unique integer polynomials $p$0 with no constant term such that

$p$1

and equivalently

$p$2

(Kellner, 5 Sep 2025).

An earlier higher-power milestone is Glaisher’s congruence, recovered by Levaillant through $p$3-adic lifting and Faulhaber polynomials: $p$4 The same method also yields a formula for $p$5 modulo $p$6 in terms of Bernoulli numbers (Levaillant, 2019).

Kellner’s second and third papers translate the $p$7-expansions into explicit congruences involving divided Bernoulli numbers and Kummer congruences, obtaining $p$8 modulo $p$9 and equivalently $(p-1)!\equiv -1\pmod p$0 modulo $(p-1)!\equiv -1\pmod p$1, then extending further to $(p-1)!\equiv -1\pmod p$2 modulo $(p-1)!\equiv -1\pmod p$3 and $(p-1)!\equiv -1\pmod p$4, and equivalently $(p-1)!\equiv -1\pmod p$5 modulo $(p-1)!\equiv -1\pmod p$6 and $(p-1)!\equiv -1\pmod p$7 (Kellner, 5 Sep 2025, Kellner, 30 Oct 2025).

Paper	Explicit reach	Main mechanism
(Levaillant, 2019)	$(p-1)!\equiv -1\pmod p$8 mod $(p-1)!\equiv -1\pmod p$9, $(n-1)!\equiv -1\pmod n$0	Hensel lifting, Faulhaber polynomials, Bernoulli numbers
(Kellner, 5 Sep 2025)	$(n-1)!\equiv -1\pmod n$1 mod $(n-1)!\equiv -1\pmod n$2, $(n-1)!\equiv -1\pmod n$3 mod $(n-1)!\equiv -1\pmod n$4	$(n-1)!\equiv -1\pmod n$5-polynomials, Bernoulli polynomials, Kummer congruences
(Kellner, 30 Oct 2025)	$(n-1)!\equiv -1\pmod n$6 mod $(n-1)!\equiv -1\pmod n$7, $(n-1)!\equiv -1\pmod n$8; $(n-1)!\equiv -1\pmod n$9 mod $n$0, $n$1	Fermat-quotient power sums, divided Bernoulli numbers

These developments show that classical Wilson is the first layer of a much richer $n$2-adic expansion, rather than an isolated mod-$n$3 coincidence.

4. Stirling-theoretic and combinatorial generalizations

A substantial generalization arises from Stirling numbers of the second kind. For positive integers $n$4 and complex $n$5, the Stirling function

$n$6

satisfies

$n$7

when $n$8, where

$n$9

With $n\ge2$0, the main arithmetic result is: $n\ge2$1 (Williams, 2016). The case $n\ge2$2 gives

$n\ge2$3

so classical Wilson is recovered exactly.

The same framework yields complementary congruences for Stirling numbers. If $n\ge2$4 is prime, then for every $n\ge2$5 and every $n\ge2$6,

$n\ge2$7

(Williams, 2016). This places Wilson’s theorem inside a broader pattern of congruences indexed by Stirling shifts.

A distinct combinatorial extension replaces $n\ge2$8 by sums of products of subsets of $n\ge2$9. If $p$00 denotes the multiset of all products of $p$01-subsets of $p$02, then for fixed $p$03 and $p$04,

$p$05

When $p$06, this reduces to Wilson’s theorem because $p$07 (Gaitanas, 2023).

Another extension interpolates among “partial factorial blocks.” For integers $p$08 and $p$09,

$p$10

if and only if $p$11, and this leads to generalized Wilson quotients

$p$12

(Morozov, 22 May 2025). This suggests a family of Wilson-type residues parameterized by the truncation point $p$13.

5. Wilson-like analogues beyond the ordinary factorial

Wilson-type congruences extend to several other factorial-like functions. For primes $p$14, Aebi and Cairns proved congruences for the superfactorial $p$15, the hyperfactorial $p$16, the double factorial $p$17, and the subfactorial $p$18 (Aebi et al., 2013). In particular, if $p$19 is prime, then

$p$20

and for odd prime $p$21,

$p$22

For all integers $p$23,

$p$24

The prime-case congruences therefore organize several factorial variants around a small set of distinguished residues.

A particularly compact relation among these functions is

$p$25

for primes $p$26 (Aebi et al., 2013). This gives a uniform residue-theoretic connection between ordinary Wilson behavior and higher factorial constructions.

There is also a finite-field analogue. For any odd prime power $p$27,

$p$28

which is the direct finite-field version of Wilson’s theorem (Bluher, 2017). Bluher further defined natural subsets of $p$29 by quadratic-character conditions and obtained explicit products for these subsets. One example is

$p$30

showing that Wilson-like product formulas persist after restricting from all nonzero field elements to Legendre-symbol slices (Bluher, 2017).

6. Relations to Giuga, Wolstenholme, and current themes

Wilson’s theorem also appears as one endpoint of an interpolation toward Giuga’s conjecture. Sauvaget introduced

$p$31

with the properties

$p$32

and

$p$33

Thus $p$34 recovers Wilson’s criterion, while $p$35 gives the Giuga-type power-sum condition (Sauvaget, 2019). The interpolation conjecture proposed there is that for every fixed $p$36,

$p$37

This suggests that Wilson’s theorem may be viewed as the first member of a broader family linking factorial and power-sum primality congruences.

The Lagrange symmetric-sum framework also connects Wilson with Wolstenholme’s theorem. Using the same elementary symmetric sums $p$38 that prove Wilson, one obtains

$p$39

which is Wolstenholme’s theorem (Hussein, 2018). In this sense Wilson’s theorem is not isolated even among classical congruences: it belongs to a cluster of prime-specific identities accessible from the same algebraic setup.

Recent work therefore treats Wilson’s theorem less as a terminal statement than as a base layer. Stirling-function constructions, Bell-polynomial supercongruences, Bernoulli-number expansions, subset-product criteria, and finite-field character products all preserve the theorem’s central theme: primality manifests itself through exceptionally rigid multiplicative congruences (Williams, 2016, Kellner, 5 Sep 2025, Bluher, 2017).