CRITERIA OF PRIMALITY DUE TO SMARANDCHE 1) Let S(n) be the Smarandache Function: S(n) is the smallest number such that S(n)! is divisible by n. Let p be an integer > 4. Then: p is prime if and only if S(p) = p. References: [1] Dumitrescu, C., "A Brief History of the Smarandache Function", , ?? [2] Smarandache, Florentin, "A Function in the Number Theory", , Fascicle 1, Vol. XVIII, 1980, pp. 79-88; reviewed in Mathematical Reviews: 83c:10008. The following four statements are derived from the Wilson theorem (p is prime iff (p-1)! is congruent to -1 (mod p)), but improve it because the factorial is reduced: 2) Let p be an integer >= 3. Then: p-1 p is prime if and only if (p-3)! is congruent to ---- (mod p). 2 References: [1] Smarandache, Florentin, "Criteria for a Positive Integer to be Prime", , Bucharest, No. 2, 1981, pp. 49-52; reviewed in Mathematical Reviews: 83a:10007. [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, Bucharest, 1996, pp. 94-98. 3) Let p be an integer > 4. Then: | p | | --- |+1 |_ 3 _| | p+1 | p is prime iff (p-4)! is congruent to (-1) | ---- | (mod p), |_ 6 _| | | where | x | means the inferior integer part of x, i.e. the smallest |_ _| integer greater than or equal to x. References: [1] Smarandache, Florentin, "Criteria for a Positive Integer to be Prime", , Bucharest, No. 2, 1981, pp. 49-52; reviewed in Mathematical Reviews: 83a:10007. [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, Bucharest, 1996, pp. 94-98. 4) Let p be an integer >= 5. Then: 2 r - 1 p is prime iff (p-5)! is congruent to rh + -------- (mod p), 24 | p | with h = | ---- | and r = p - 24h, |_ 24 _| | | where | x | means the inferior integer part of x, i.e. the smallest |_ _| integer greater than or equal to x. References: [1] Smarandache, Florentin, "Criteria for a Positive Integer to be Prime", , Bucharest, No. 2, 1981, pp. 49-52; reviewed in Mathematical Reviews: 83a:10007. [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, Bucharest, 1996, pp. 94-98. 5) Let p = (k-1)!h + 1 be a positive integer, k > 2, h natural number. Then: t p is prime iff (p-k)! is congruent to (-1) h (mod p), | p | with t = h + | ---- | + 1, |_ h _| | | where | x | means the inferior integer part of x, i.e. the smallest |_ _| integer greater than or equal to x. References: [1] Smarandache, Florentin, "Criteria for a Positive Integer to be Prime", , Bucharest, No. 2, 1981, pp. 49-52; reviewed in Mathematical Reviews: 83a:10007. [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, Bucharest, 1996, pp. 94-98. 6) Let 1 <= k <= p be integers. Then: p is prime if and only if (p-k)!(k-1)! is congruent to (-1)^k. References: [1] Smarandache, Florentin, "Criteria for a Positive Integer to be Prime", , Bucharest, No. 2, 1981, pp. 49-52; reviewed in Mathematical Reviews: 83a:10007. [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, Bucharest, 1996, pp. 94-98.