EULER'S TOTIENT FUNCTION AND CONGRUENCE THEOREM GENERALIZED BY SMARANDACHE Let a, m be integers, m different from 0. Then: phi(m ) + s s s a is congruent to a (mod m), where phi(x) is Euler's Totient Function, and s and m are obtained through the algorithm: s ___ | a = a d ; (a , m ) = 1; | 0 0 0 0 (0) | | m = m d ; d different from 1; | 0 0 0 --- ___ 1 1 | d = d d ; (d , m ) = 1; | 0 0 1 0 1 (1) | | m = m d ; d different from 1; | 0 1 1 1 --- .................................................... ___ 1 1 | d = d d ; (d , m ) = 1; | s-2 s-2 s-1 s-2 s-1 (s-1)| | m = m d ; d different from 1; | s-2 s-1 s-1 s-1 --- ___ 1 1 | d = d d ; (d , m ) = 1; | s-1 s-1 s s-1 s (s) | | m = m d ; d = 1. | s-1 s s s --- Therefore it is not necessary for a and m to be coprime. 25604 Example: 6 is congruent to ? (mod 105765). It is not possible to use Fermat's or Euler's theorems, but the Smarandache Congruence Theorem works: d = (6, 105765) = 3 0 m = 105765/3 = 35255 0 i = 0 3 is different from 1 thus i = 0+1 = 1 d = (3, 35255) = 1 1 m = 35255/1 = 35255. 1 phi(35255)+1 1 Therefore 6 is congruent to 6 (mod 105765) 25604 4 whence 6 is congruent 6 (mod 105765). References: [1] Porubsky, Stefan, "On Smarandache's Form of the Individual Fermat- Euler Theorem", , Vol. 8, No. 1-2-3, Fall 1997, pp. 5-20, ISSN 1084-2810. [2] Porubsky, Stefan, "On Smarandache's Form of the Individual Fermat- Euler Theorem", , University of Craiova, August 21-24, 1997, pp. 163-178, ISBN 1-879585-58-8. [3] Smarandache, Florentin, "Une generalization de theoreme d'Euler" (French), , Seria C, Vol. XXIII, 1981, pp. 7-12, reviewed in Mathematical Reviews: 84j:10006. [4] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, Bucharest, 1996, pp. 182-191.