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Wilson Theorem Calculator

Work out wilson theorem instantly with clear inputs, formula shown and shareable results.

About Wilson Theorem Calculator

Wilson's theorem gives an exact primality criterion: p is prime if and only if (p − 1)! ≡ −1 (mod p). It is beautiful but useless for large p, since computing the factorial is far slower than trial division.

Formula

Wilson's theorem

(p − 1)! ≡ −1 (mod p) ⇔ p is prime

Worked example

With the default values below, Wilson Theorem Calculator returns (p−1)! mod p of 12. Change any input above and every figure here is recalculated in your browser.

Inputs used

Number to testaccepts 2–500
13
Also testaccepts 2–500
15

Results

(p−1)! mod p
12
Interpretation
13 is prime — (p−1)! ≡ −1 (mod p)
Value expected for a prime (p − 1)
12
(p−1)! mod p for the second number
0

Step by step

  1. 1Values used
  2. 2Wilson's theorem
  3. 3(p−1)! mod p= 12
  4. 4Interpretation= 13 is prime — (p−1)! ≡ −1 (mod p)
  5. 5Value expected for a prime (p − 1)= 12
  6. 6(p−1)! mod p for the second number= 0

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Check 13.

12! mod 13 = 12, which is −1 mod 13, confirming 13 is prime.

Why does it fail for composites?

For composite p above 4, the factors of p all appear among 1 to p − 1, so the factorial is divisible by p and the residue is 0.

Related Calculators

Wilson Theorem Calculator – Calculate (p−1)! mod p