Utilizing LTE to Analyze Recursive Patterns in Prime Divisibility of Polynomial Powers

This paper explores the application of the Lifting The Exponent (LTE) theorem to uncover recursive patterns in prime divisibility within polynomial sequences of the form and , where a, b, and n are integers and p is a prime. The LTE theorem provides a systematic method for calculating p-adic valuations, revealing predictable patterns in divisibility by powers of p as n increases. By applying LTE, we derive concise recursive relations that describe the highest powers of p dividing each term in these sequences, without direct computation of large powers. We present case studies illustrating LTE’s utility in establishing prime power divisibility across exponential terms, enabling efficient analysis of factorization and periodicity in integer sequences. This work highlights LTE’s role in advancing number-theoretic methods for modular arithmetic, cryptography, and the classification of prime-divisibility properties in polynomial expressions, underscoring its value as a foundational tool in modern mathematical analysis.