Wilson's Theorem Primality Checker

Wilson's Theorem Primality Checker | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: text, select
Output type: text
Sample coverage: 4
API ready: Yes

Overview

The Wilson's Theorem Primality Checker evaluates whether an integer n is prime by computing the factorial residue (n−1)! modulo n. It tests the classical congruence (n−1)! ≡ −1 (mod n), cross-checks the result using deterministic Miller–Rabin, evaluates the composite factorial law, and optionally outputs running modular products.

When to use

Verifying primality for integers up to 10⁷ using exact Wilson's theorem modular reduction.
Studying step-by-step modular residue sequences k! mod n for educational or number theory demonstrations.
Demonstrating the composite factorial law where (n−1)! ≡ 0 (mod n) for composite numbers greater than 4.

How it works

1Enter an integer n within the supported range (2 ≤ n ≤ 10⁷) and choose the desired output detail level.
2The tool iteratively multiplies k from 1 to n−1 modulo n to compute the exact residue of (n−1)! mod n.
3It checks if the residue equals n − 1 (representing −1 mod n) and applies deterministic Miller–Rabin to provide an independent cross-check.

Use cases

Number theory students inspecting the sequence of running products k! mod n to observe modular multiplicative inverses.
Mathematics educators verifying primality test proofs with deterministic cross-checking.
Developers validating modular arithmetic implementations against verified mathematical theorems.

Examples

1. Verifying a prime with running products

Mathematics Student

Background

A student is analyzing how modular inverses pair up during the factorial accumulation modulo a prime.

Problem

Check if 13 is prime using Wilson's theorem and examine the step-by-step product trace.

How to use

Enter 13 into the Number n field and select 'Also show running products (n ≤ 200)' in Output detail.

n = 13, detail = trace

Outcome

Shows 12! mod 13 = 12, confirms Wilson's condition holds, lists all 12 intermediate products, and confirms primality via Miller–Rabin.

2. Checking a composite number under the factorial law

Math Educator

Background

A teacher is preparing lecture materials demonstrating why composites collapse to 0 modulo n.

Problem

Evaluate n = 9 to show that Wilson's congruence fails and the composite factorial law holds.

How to use

Enter 9 into the Number n field and select 'Verdict and cross-check'.

FAQ

What is Wilson's theorem?

Wilson's theorem states that a natural number n > 1 is prime if and only if (n−1)! ≡ −1 (mod n).

Why does the tool limit running products to n ≤ 200?

Generating and displaying the full list of intermediate residues k! mod n is restricted to n ≤ 200 to keep the output readable and performant.

What is the composite factorial law?

For every composite number n > 4, (n−1)! ≡ 0 (mod n) because its factors appear before reaching n−1.

Why is Wilson's theorem rarely used for massive cryptographic primes?

Computing (n−1)! takes O(n) modular operations, which becomes computationally infeasible for very large numbers compared to O(log n) tests like Miller–Rabin.

What is the only composite exception where (n−1)! is not 0 mod n?

The only composite exception is n = 4, where (4−1)! = 3! = 6 ≡ 2 (mod 4).

n = 9, detail = verdict

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Verifying a prime with running products

2. Checking a composite number under the factorial law

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