Wilson prime checker

The Wilson prime checker answers a precise modular question from elementary number theory: given a prime p, does p squared divide (p minus one) factorial plus one? Equivalently, is (p−1)! congruent to −1 modulo p²? Wilson's theorem already guarantees that every prime satisfies the same congruence modulo p, so the Wilson-prime condition is a genuine strengthening of the modulus rather than a new kind of primality test. Supply a prime p and the tool returns p, p squared, the exact remainder of (p−1)! The classic small Wilson primes are five and thirteen; five hundred sixty-three is the only other known Wilson prime and still sits comfortably inside the supported range. The engine multiplies successive integers under a BigInt modulus, never samples randomness, never reads the clock, and never leaves the process for a network call.

What a Wilson prime is and what you see

A Wilson prime is a prime number p such that p squared divides (p minus one) factorial plus one. In modular language the same statement reads (p−1)! ≡ −1 (mod p²).

Wilson's theorem says that for every prime p one already has (p−1)! ≡ −1 (mod p), so the factorial plus one is always divisible by p; the Wilson condition asks whether that divisibility lifts from modulus p all the way to modulus p². The only Wilson primes known in the literature are five, thirteen, and five hundred sixty-three; exhaustive searches far beyond one million have found no further examples, which makes the three-member list a favorite illustration of how rare modular liftings can be.

≡ -1 (mod p^2). If you only need ordinary primality, use a dedicated prime checker; this page answers the Wilson modular question with an explicit remainder.

How inputs are read and how the modular factorial stays exact

Optional leading plus signs are allowed for positive values; embedded spaces, hex prefixes, and exponential notation are rejected so a bad cell does not become a wrong flag. Internally the engine first proves primality by trial division up through the square root of p, which is cheap at this scale and fully deterministic.

It then walks the product two times three times four up through p minus one, reducing modulo p squared after every multiply with BigInt arithmetic so intermediate values never overflow a fixed machine word and never drift into floating-point territory. The final remainder of that product plus one modulo p squared is converted back to a safe JavaScript number—p squared is at most one trillion for the allowed range—and compared with zero.

Where Wilson checks fit in study, research demos, and code

Students meet Wilson primes when studying Wilson's theorem, modular arithmetic, and rare prime specializations alongside Sophie Germain primes, safeprimes, and Wilson quotients. Teachers can generate answer keys for worksheets without trusting opaque calculator apps that hide the modular product step or silently accept composites. Developers use the same contract to validate sample data for number-theory libraries, to gate property tests on random prime generators, or to pre-check that a demo seed is one of the three classic Wilson primes before a live presentation.

Contest practice often needs a fast offline source of Wilson answers so solutions can be checked without downloading OEIS tables or reimplementing BigInt factorial modular reduction under time pressure. Pair it with an ordinary prime checker when you need only primality, with a twin-prime checker when you need consecutive odd primes, and with modular-exponentiation tools when you need general pow-mod rather than a Wilson-specific factorial product.