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arXiv · 1810.11456

Primitive Indexes, Zsigmondy Numbers, and Primoverization

Tejas Rao
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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.

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Tejas Rao. 2018-10-25. Primitive Indexes, Zsigmondy Numbers, and Primoverization. https://arxiv.org/abs/1810.11456

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