A Study On The Properties Of Fermat Quotients

Fermat quotients arose from the study of the first case of Fermat’s Last Theorem,and many authors have done a lot of research on it.Fermat quotients have numerous applications in computational and algebraic number theory,and plays an important role in generating pseudorandom sequences.In this paper,we continue to study the properties of Fermat quotients by means of number theory and algebraic tools,and the main results are as follows.Firstly,we studied the relation between the non-negative least residue and Fermat quotient of an integer.We study the mean value distribution of the difference between the non-negative least residue rp(n)and Fermat quotient qp(n)of an integer n by using the properties of the residue systems,trigonometric sum,character sum and binomial expansion,and give an identity.Secondly,we constructed a few pseudorandom subsets by using Fermat quo-tient.Based on the set defined by Dartyge,Mosaki and Sarkozy new pseudo-random subsets are constructed by combining Fermat quotient.We studied the well-distribution measure and the correlation measure of the new pseudorandom subsets by using the properties of exponential sum,character sum,trigonometric sum and primitive roots.