HYBRIDIZING HENSEL’S LEMMA, FUNDAMENTAL THEOREM OF ARITHMETIC, AND CHINESE REMAINDER THEOREM FOR SOLVING POLYNOMIAL CONGRUENCES

Abstract

Polynomial congruence can be solved by applying Hensel’s Lemma. However, Hensel’s Lemma itself does not apply to solving generalized polynomial congruences. The purpose of this research is to determine the recursive formula for the solution of polynomial congruence modulo prime numbers and to construct a general solution algorithm of polynomial congruence modulo arbitrary positive integers. Unlike previous studies, this research proposes the recursive hybrid algorithm combining Hensel’s Lemma, the Fundamental Theorem of Arithmetic, and the Chinese Remainder Theorem, highlighting the originality of the approach in extending its application beyond prime power moduli. The result of this research is the form of a recursive formula for the solution of polynomial congruence modulo prime numbers and the algorithm for solving polynomial congruence modulo arbitrary positive integers using the combination of Hensel’s Lemma, Fundamental Theorem of Arithmetic, and Chinese Remainder Theorem. The results of this research contribute to the development of mathematical methods, especially in the field of number theory. However, the applicability of the recursive formula is limited to cases where the conditions of Hensel’s Lemma are satisfied, that is, when a solution of the polynomial modulo a prime is such that the polynomial equals zero while its derivative does not equal zero modulo the same prime. Extending the method to situations where this condition fails remains a subject for future research.