Primitive Indexes, Zsigmondy Numbers, and Primoverization

TL;DR

This paper introduces a new concept of primitive index in sequences, derives formulas relating primitive indexes and multiplicative orders, and explores implications for Zsigmondy numbers, pseudoprimes, and special prime classes.

Contribution

It provides novel formulas for primitive indexes in specific sequences and extends existing lemmas, offering new insights into prime factorization and pseudoprime classification.

Findings

Derived a formula linking primitive indexes of composite numbers to prime factors.

Expanded the lifting the exponent lemma for specific sequences.

Proved Wagstaff numbers are either overpseudoprime or prime.

Abstract

We define a primitive index of an integer in a sequence to be the index of the term with the integer as a primitive divisor. For the sequences $k^u+h^u$ and $k^u-h^u$, we discern a formula to find the primitive indexes of any composite number given the primitive indexes of its prime factors. We show how this formula reduces to a formula relating the multiplicative order of $k$ modulo $N$ to that of its prime factors. We then introduce immediate consequences of the formula: certain sequences which yield the same primitive indexes for numbers with the same unique prime factors, an expansion of the lifting the exponent lemma for $v_2(k^n+h^n)$, a simple formula to find any Zsigmondy number, a note on a certain class of pseudoprimes titled overpseudoprime, and a proof that numbers such as Wagstaff numbers are either overpseudoprime or prime.