"Application of Fermat’s Little Theorem in Congruence Relation Modulo n"

Abstract

According to Fermat's little theorem, for any p is a prime integer and 𝑔𝑐𝑑(𝑥, 𝑝) = 1, then the congruence 𝑥 𝑝−1 ≡ 1(𝑚𝑜𝑑 𝑛 )is true, if we remove the restriction that 𝑔𝑐𝑑(𝑥, 𝑝) = 1, we may declare𝑥 𝑝−1 ≡ 𝑥(𝑚𝑜𝑑 𝑝). For every integer x. Euler extended Fermat's Theorem as follows: if 𝑔𝑐𝑑(𝑥, 𝑝) = 1,then,where 𝑥 𝜙(𝑛) ≡ 1(𝑚𝑜𝑑 𝑛).𝜙 is Euler's phi-function. Euler's theorem cannot be implemented for any every integers x in the same manners as Fermat’s theorem works; that is, the congruence 𝑥 𝜙(𝑛)+1 ≡ 𝑥(𝑚𝑜𝑑 𝑛) is not always true. In this paper, we discussed the validation of congruence 𝑥 𝜙(𝑛)+1 ≡ 𝑥(𝑚𝑜𝑑 𝑛).