The week's pick
Reseach Article
A Geometric Construction Involving Wilson’s Theorem
| International Journal of Computer Applications |
| Foundation of Computer Science (FCS), NY, USA |
| Volume 175 - Number 1 |
| Year of Publication: 2017 |
| Authors: Kenneth J. Prevot |
10.5120/ijca2017915463
|
Kenneth J. Prevot . A Geometric Construction Involving Wilson’s Theorem. International Journal of Computer Applications. 175, 1 ( Oct 2017), 6-8. DOI=10.5120/ijca2017915463
Abstract
A long standing result in number theory is Wilson’s Theorem, which states that n is a prime number if and only if (n – 1)! ≡ (-1) mod n. One motivation for this study is to detect some algebraic congruence relations which naturally arise in this number theoretic context, strictly through geometric constructions. Some examples of such congruence relations are presented. Namely, than n is an odd prime if and only if (n – 2)! – n(n – 3)/2 ≡ 1 mod (n2 -2n). Also if n is an odd prime, one has (n – 2)((n – 1)!)+(n –1) ≡1 mod (n2 – 2n).
References
- Rosen, K. H., Elementary Number Theory and its applications, 4th edition, Addison Wesley Longman, Reading, Massachusetts, 1999.
- Gabai, David, “Foliations and topology of 3-manifolds,” J. Diff. Georm. 18 (1983) no. 3, 445-503.
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