Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques

M Khalid Mahmood,Farooq Ahmad

doi:10.1155/2014/575064

Abstract

In this paper we put forward a family of algorithms for lifting solutions of a polynomial congruencemod pto polynomial congruencemod pk. For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the formaxn≡b(mod pk),k≥1,wherea,b,andn>0are integers which are not divisible by an odd primep. It is shown that the algorithms suggested in this paper drastically reduce the complexity for such computations to a logarithmic scale. The efficacy of the proposed technique for solving negative exponent equations of the formax-n≡b(mod pk)has also been addressed.

Highlights

Introduction and PreliminariesThe scope of congruence in number theory is of vital importance
This research work addresses some iterative methods for solving polynomial congruences of the form axn ≡ b(mod pk), k ≥ 1 where a, b, and n > 0 are integers which are not divisible by an odd prime p
Basic family of iteration functions denoted by Bm(x) is a well-known class of iterative algorithms of order m for solving a nonlinear equation in one variable

Summary

Introduction and Preliminaries

The scope of congruence in number theory is of vital importance. The use of iterative methods for solving nonlinear equations has become a valuable device for numerical analysts. The root-finding recursive techniques have been discussed in [1,2,3,4] to get the inverse of numbers modulo prime powers, which is the motivation of the proposed research work In this piece of work, we use higher order iterative methods with a particular focus on Householder’s and Basic Family of Iteration Functions (for detail, see [5,6,7]) in order to find solutions of polynomial congruences of the form axn ≡ b(mod pk), k ≥ 1. By Theorem 2, there is a unique integer solution s2 t from the coofnfg(rxu)en≡ce0(fm󸀠(o3d)t7≡2).(T−off(i3n)d/7s)2(,mwoedfin7)d This gives 54t ≡ (−49/7)(mod 7) or t ≡ 0(mod 7). In the underlying paper we solve the polynomial congruence with higher modulo by means of algorithms developed using root-finding iterative methods. Notations used in this paper are standard and we follow [1,2,3, 10, 11]

A Solution of Congruences Using Newton’s Method

Third Order Iterative Methods

Higher Order Iterative Families

Numerical Examples

Conclusion

Methods