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.