Existence and Distribution of Solutions of a^x = b modulo p^n

Initial objective of this dissertation is to study the existence of the solutions of the congruence a^x = b^y (mod p^n) and distribution of solutions (x, y) as n varies in natural numbers, where a and b are integers coprime to prime p. We observe that as n tends to infinity, solutions take the form of p-adic integers. This motivates us, to study the existence of the solutions of equation a^x = b in p-adic integers. The relevant case is when a and b are units in p-adic integers. If the solution exists we try to find it out. We resolve the case of a, b in U_1 completely. A necessary and sufficient condition for the existence of the solution of a^x = b where a, b are elements of U_1, is `valuation of a-1 is smaller than the valuation of b-1'. In this case, if the solution exists then it is given by log b / log a. In the other case, where a and b are p-adic units but not elements of U_1, we give the criteria for the existence of the solution of a^x = b. Write a and b as product of Teichmuller unit and an element of U_1. Suppose a= a_1 a_2 and b = b_1 b_2 where a_1, b_1 are Teichmuller units and a_2, b_2 are in U_1. Then the solution of a^x = b exists if b_1 belongs to the group generated by a_1 and valuation of a_2 - 1 is smaller than the valuation of b_2 -1.