Primitive Indexes, Zsigmondy Numbers, and Primoverization

2018/10/25 by Rao, Tejas
#11A99 (secondary) #11B99 (primary) 11A07 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1810.11456

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 v₂(kⁿ+hⁿ), 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.

Related

• The index and its prime divisors
• Automatic Bounds on Constant Term Sequences Modulo Primes
• The use of the invariant's properties in the primality test and prime search
• Sequences of Numbers Involved in Unsolved Problems
• A note on primes with prime indices