IJPAM: Volume 19, No. 4 (2005)

ON THE CONGRUENCES $a^{\phi (n)+L}\equiv a(\,\text{\bf mod}\,n)$
AND $a^{n+L}\equiv a(\,\text{\bf mod}\,n)$

A.N. El-Kassar
Department of Mathematics
Faculty of Science
Beirut Arab University Tarik El-Jedidah
P.O. Box 11-5020, Beirut, LEBANON
e-mail: ak1@bau.edu.lb

Abstract.Fermat's Theorem states that if is prime, then $a^{n}\equiv a(\text{\rm mod}% n)$ holds for every integer Euler's Theorem states that the congruence $% a^{\phi (n)}\equiv 1(\text{\rm mod}n),$ and hence $a^{\phi (n)+1}\equiv a(\text{\rm % mod}n),$ holds for every integer relatively prime to , where $\phi (n)$ is Euler phi- function. If the condition that and are relatively prime is dropped, then the last congruence holds for every integer iff is a product of distinct primes. The congruences $% a^{\phi (n)+L}\equiv a(\text{\rm mod}n)$ and $a^{n+L}\equiv a(\text{\rm mod}n)$ are examined. Given the values of for which the congruence $a^{\phi (n)+L}\equiv a(\text{\rm mod}n)$ ( $a^{n+L}\equiv a(\text{\rm mod}n)$ ) holds for every integer are characterized. In addition, properties of solutions are studied. The two congruences are extended to finite commutative rings with identity. For a fixed a characterization of all finite commutative rings with identity for which $a^{\varphi (R)+L}=a$ ( $a^{r+L}=a$ ) for every $a\in R$ is given where is the order of and $\varphi (R)$ is the order of its group of units.