Wilson Quotient in Number Theory

• Wilson quotient is defined as ((p-1)! + 1)/p for primes, serving as a correction term to Wilson’s theorem and offering insights into p-adic structures.
• It distinguishes Wilson primes and links to Fermat quotients, Bell numbers, and left factorials, revealing deep arithmetic and combinatorial relationships.
• Recent studies extend Wilson’s theorem to higher powers using recursive polynomial techniques and divided Bernoulli numbers to achieve supercongruence results.

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 (Howard, 8 Apr 2026, Kellner, 5 Sep 2025, Morozov, 22 May 2025).

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 $p\ge 3$0 to obtain $p\ge 3$1, since $p\ge 3$2 is even. An equivalent derivation uses the multiplicative group $p\ge 3$3 (Howard, 8 Apr 2026).

Because $p\ge 3$4, the quotient

$p\ge 3$5

lies in $p\ge 3$6. Equivalently,

$p\ge 3$7

This makes $p\ge 3$8 the first-order correction term to Wilson’s congruence in the $p\ge 3$9-adic direction. The quantity is often studied modulo powers of $(p-1)!\equiv -1\pmod p$0, not only modulo $(p-1)!\equiv -1\pmod p$1 itself, because higher congruence data encode substantially finer information than the classical statement alone (Howard, 8 Apr 2026, Kellner, 30 Oct 2025).

A useful characterization is that

$(p-1)!\equiv -1\pmod p$2

if and only if $(p-1)!\equiv -1\pmod p$3 is a Wilson prime. This is precisely the condition $(p-1)!\equiv -1\pmod p$4 (Howard, 8 Apr 2026).

2. Wilson primes, elementary examples, and residue-level behavior

A prime $(p-1)!\equiv -1\pmod p$5 is called a Wilson prime if

$(p-1)!\equiv -1\pmod p$6

Only three are known:

$(p-1)!\equiv -1\pmod p$7

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 (Howard, 8 Apr 2026).

For small primes, the quotient is explicit. When $(p-1)!\equiv -1\pmod p$8,

$(p-1)!\equiv -1\pmod p$9

When $\mathbb Z$0,

$\mathbb Z$1

so $\mathbb Z$2 is a Wilson prime. When $\mathbb Z$3,

$\mathbb Z$4

These examples already show that integrality of $\mathbb Z$5 is universal for primes, whereas vanishing modulo $\mathbb Z$6 is exceptional (Howard, 8 Apr 2026).

The residue class of $\mathbb Z$7 modulo $\mathbb Z$8 is also tied to classical congruences of Lerch type. In particular, later higher-power work recovers

$\mathbb Z$9

where

$p$0

Thus the mod-$p$1 Wilson quotient is expressible as a power sum of Fermat quotients, linking Wilson’s theorem to a separate but closely related arithmetic framework (Kellner, 5 Sep 2025).

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$2, the left factorial is

$p$3

while the Bell numbers $p$4 count set partitions and satisfy the exponential generating function

$p$5

Gertsch’s congruence asserts that for every odd prime $p$6,

$p$7

According to the cited account, this follows from combining Touchard’s congruences for Bell polynomials with the congruence $p$8 (Howard, 8 Apr 2026).

Within this setting, the paper "The Wilson Quotient, Left Factorials, Bell Numbers, and the Kurepa–Bell–Wilson Congruence" formulates the Kurepa–Bell–Wilson congruence:

$p$9

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 (Howard, 8 Apr 2026).

The same work defines the Gertsch quotient

$p$0

From the KBW congruence one obtains

$p$1

Hence $p$2 is an integer lift of the residue of $p$3. The paper further interprets this lift in the “poor man’s adele ring”

$p$4

via the natural diagonal embedding of $p$5 (Howard, 8 Apr 2026).

The small-prime instances given in the source illustrate the congruence concretely. For $p$6, one has $p$7 and $p$8. For $p$9, one has $(p-1)!\equiv -1\pmod p.$0 and $(p-1)!\equiv -1\pmod p.$1, hence both are congruent modulo $(p-1)!\equiv -1\pmod p.$2. For $(p-1)!\equiv -1\pmod p.$3, one has $(p-1)!\equiv -1\pmod p.$4 and $(p-1)!\equiv -1\pmod p.$5, again with

$(p-1)!\equiv -1\pmod p.$6

The same source records that no prime $(p-1)!\equiv -1\pmod p.$7 is known for which $(p-1)!\equiv -1\pmod p.$8 (Howard, 8 Apr 2026).

4. Supercongruences and higher prime-power expansions

A major contemporary development is the extension of Wilson’s theorem from modulus $(p-1)!\equiv -1\pmod p.$9 to arbitrary higher powers $x^{p-1}-1$0. For an odd prime $x^{p-1}-1$1 with $x^{p-1}-1$2, Kellner proves that there exist unique multivariate polynomials

$x^{p-1}-1$3

each with no constant term, such that

$x^{p-1}-1$4

and equivalently

$x^{p-1}-1$5

For $x^{p-1}-1$6 this reproduces Lerch’s congruence

$x^{p-1}-1$7

(Kellner, 5 Sep 2025).

The construction uses partial Bell polynomials and Newton’s identities. The elementary symmetric sums of the family $x^{p-1}-1$8 are rewritten in terms of the power sums $x^{p-1}-1$9, and the key factorization

$\mathbb Z/p\mathbb Z$0

then ties the resulting symmetric-polynomial data to powers of $\mathbb Z/p\mathbb Z$1. The recursion for the $\mathbb Z/p\mathbb Z$2 is uniform in $\mathbb Z/p\mathbb Z$3, and the low-degree cases are explicitly

$\mathbb Z/p\mathbb Z$4

$\mathbb Z/p\mathbb Z$5

(Kellner, 5 Sep 2025).

A further refinement computes the Wilson quotient modulo $\mathbb Z/p\mathbb Z$6 and $\mathbb Z/p\mathbb Z$7, and equivalently $\mathbb Z/p\mathbb Z$8 modulo $\mathbb Z/p\mathbb Z$9 and $x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$0, respectively. In that framework one introduces the divided Bernoulli numbers

$x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$1

which lie in $x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$2 and satisfy the usual Kummer congruences. For $x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$3,

$x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$4

and for $x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$5,

$x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$6

The coefficients $x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$7 admit closed forms in the divided Bernoulli numbers; for instance,

$x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$8

in the modulus-$x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod p,$9 theorem (Kellner, 30 Oct 2025).

The higher-power study also records a regular pattern in the reductions $p\ge 3$00: the subsequence of pure powers $p\ge 3$01 appears, along with analogous universal terms in $p\ge 3$02 and $p\ge 3$03. The paper formulates the conjectural regularity that for all $p\ge 3$04, the reduction $p\ge 3$05 is a $p\ge 3$06-linear combination of monomials

$p\ge 3$07

with rational coefficients depending only on $p\ge 3$08 (Kellner, 30 Oct 2025).

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 $p\ge 3$09 and any integer $p\ge 3$10 with $p\ge 3$11,

$p\ge 3$12

When $p\ge 3$13, this recovers Wilson’s theorem (Morozov, 22 May 2025).

For a prime $p\ge 3$14, Morozov defines

$p\ge 3$15

The case $p\ge 3$16 is exactly the Wilson quotient:

$p\ge 3$17

Thus the classical quotient sits inside a finite family of quotients indexed by the position of the factorial split (Morozov, 22 May 2025).

This family satisfies closed-form summation identities. For

$p\ge 3$18

one has

$p\ge 3$19

At the residue level, Morozov proves Lehmer-type congruences:

$p\ge 3$20

where $p\ge 3$21 are Bernoulli numbers and $p\ge 3$22 are harmonic numbers. The paper also gives the $p\ge 3$23-level congruence

$p\ge 3$24

for the related notation $p\ge 3$25 (Morozov, 22 May 2025).

The sequence additionally admits an exponential generating function. For fixed $p\ge 3$26,

$p\ge 3$27

and

$p\ge 3$28

Since $p\ge 3$29, the same EGF describes the sequence $p\ge 3$30 (Morozov, 22 May 2025).

6. Open questions and conceptual significance

The classical open problem concerns Wilson primes: whether the condition $p\ge 3$31 is satisfied for infinitely many primes, or only finitely many. The available data record only the three known examples $p\ge 3$32 (Howard, 8 Apr 2026).

A parallel question arises from the Gertsch quotient. Since

$p\ge 3$33

one may ask whether there exist primes $p\ge 3$34 with $p\ge 3$35. The cited source states that no such prime is known, and that a search up to large bounds has turned up no example. A plausible implication is that the Bell-factorial lift encoded by $p\ge 3$36 is at least as sparse in its vanishing behavior as the ordinary Wilson quotient, though the data stop short of a structural explanation (Howard, 8 Apr 2026).

The higher-power theory indicates that the Wilson quotient is not merely a residue-level invariant but part of a full $p\ge 3$37-adic expansion controlled by recursively computable polynomials in Fermat-quotient power sums and, at deeper levels, by divided Bernoulli data (Kellner, 5 Sep 2025, Kellner, 30 Oct 2025). This suggests that Wilson’s theorem fits naturally into a supercongruence framework alongside Bell polynomials, Newton identities, and Kummer-type congruences.

From the conceptual viewpoint emphasized in the KBW setting, the quantity $p\ge 3$38 in $p\ge 3$39 decomposes into a Bell-part and a Wilson-part. The source explicitly notes that the interplay among $p\ge 3$40, $p\ge 3$41, and $p\ge 3$42 suggests new analogues of Euler’s constant and multiple-zeta-value phenomena in the finite-modulus setting, and asks how this decomposition extends to higher factorial sums or other combinatorial sequences (Howard, 8 Apr 2026). Within that perspective, the Wilson quotient functions simultaneously as a classical correction term, a modular residue, a $p\ge 3$43-adic coefficient, and a bridge between factorial arithmetic and combinatorial enumeration.