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

TL;DR

This paper investigates the existence and distribution of solutions to the congruence a^x = b modulo p^n, exploring solutions in p-adic integers and providing explicit criteria for their existence based on valuations and units.

Contribution

It offers a complete characterization of solutions for a^x = b in p-adic units, including necessary and sufficient conditions and explicit solution formulas.

Findings

Solutions tend to p-adic integers as n increases.

Explicit criteria for solution existence based on valuations.

Solution formula involving logarithms when conditions are met.

Abstract

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.