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In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n.
26 kwi 2024 · Wilson’s theorem states that any positive integer, n (> 1), is a prime number if and only if (n – 1)! ≡ -1 (mod n), which means: If (n – 1)! ≡ -1 (mod n), then n is prime; If n is prime, then (n – 1)! ≡ -1 (mod n), the converse; It is used in mathematical calculations in elementary number theory involving (n – 1)!.
Wilson's theorem states that . a positive integer \( n > 1 \) is a prime if and only if \( (n-1)! \equiv -1 \pmod {n} \). In other words, \( (n-1)! \) is 1 less than a multiple of \(n\). This is useful in evaluating computations of \( (n-1)! \), especially in Olympiad number theory problems.
In number theory, Wilson's Theorem states that if integer , then is divisible by if and only if is prime. It was stated by John Wilson. The French mathematician Lagrange proved it in 1771.
Wilson’s Theorem:If p is a prime, then (p 1)! 1 (mod p). Let n 2 be a positive integer. Then (n 1)! 1 (mod n)if and only if n is a prime. More general, If r 1;:::;r p 1 is a reduced system of residues modulo p, then r 1r 2 r p 1 1 (mod p). Gexin Yu gyu@wm.edu Math 412: Number Theory Lecture 7: Wilson’s theorem
26 lis 2024 · Unlike Fermat's little theorem, Wilson's theorem is both necessary and sufficient for primality. For a composite number, (n-1)!=0 (mod n) except when n=4. A corollary to the theorem states that iff a prime p is of the form 4k+1, then [(2k)!]^2=-1 (mod p)....
17 lis 2024 · The proof of Wilson's Theorem was attributed to John Wilson by Edward Waring in his $1770$ edition of Meditationes Algebraicae. It was first stated by Ibn al-Haytham ("Alhazen") . It appears also to have been known to Gottfried Leibniz in $1682$ or $1683$ (accounts differ).
