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Wilson's theorem - mklabgr - 07-28-2026
Wilson's theorem
Summary
Wilson's theorem is a fundamental result in number theory stating that a natural number $n > 1$ is a prime number if and only if the product of all positive integers less than $n$ is one less than a multiple of $n$—expressed in modular arithmetic as $(n-1)! \equiv -1 \pmod n$.
First stated by Persian mathematician Ibn al-Haytham around 1000 AD and later named after John Wilson (with its first published proof by Joseph-Louis Lagrange in 1771), the theorem provides a theoretical condition for primality, though it is practically useless for testing large prime numbers due to the steep computational complexity of calculating large factorials.
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