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# Smarandache Criteria of Primality

IP.com Disclosure Number: IPCOM000128826D
Original Publication Date: 1999-Sep-11
Included in the Prior Art Database: 2005-Sep-19
Document File: 2 page(s) / 13K

## Publishing Venue

Software Patent Institute

## Related People

M. L. Perez: AUTHOR [+2]

## Abstract

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. 2) Let p be an integer = 3... and 3) Let p be an integer 4...

This text was extracted from a PDF file.
This is the abbreviated version, containing approximately 61% of the total text.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Smarandache Criteria of Primality

edited by M. L. Perez - ASCHII file

SMARANDACHE CRITERIA OF PRIMALITY

edited by M. L. Perez Erhus Univ. Press Rehoboth, Box 141, NM 87322, USA

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",

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). References; [1] Smarandache,
Florentin, "Criteria for a Positive Integer to be Prime",

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",

4) Let p be an integer = 5.

M.L. Perez Page 1 Sep 11, 1999

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Smarandache Criteria of Primality

Then: 2 r...