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Counting Frobenius Pseudoprimes

Andrew Fiori, Hiva Gheisari

Abstract

We generalize the work of Erdos-Pomerance and Fiori-Shallue on counting Frobenius pseudoprimes from the cases of degree one and two respectively to arbitrary degree. More specifically we provide formulas for counting the number of false witnesses for a number $n$ with respect to Grantham's Frobenius primality test. We also provide conditional assymptotic lower bounds on the average number of Frobenius pseudoprimes and assymptotic upper bounds on the same.

Counting Frobenius Pseudoprimes

Abstract

We generalize the work of Erdos-Pomerance and Fiori-Shallue on counting Frobenius pseudoprimes from the cases of degree one and two respectively to arbitrary degree. More specifically we provide formulas for counting the number of false witnesses for a number with respect to Grantham's Frobenius primality test. We also provide conditional assymptotic lower bounds on the average number of Frobenius pseudoprimes and assymptotic upper bounds on the same.

Paper Structure

This paper contains 9 sections, 35 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Grantham's Definition and Reinterpretations
  3. Computing $L_d(n)$
  4. Lower Bounds
  5. Number Theory Background
  6. Construction
  7. Upper Bounds
  8. Numbers with large prime factors
  9. Numbers with many prime factors

Key Result

Proposition 2.1

Let $g_1(x), g_2(x)$ be monic polynomials in $\mathbb{Z}[x]$. Then if and only if for all $p^r || n$ and the $f(x) \pmod{p^r}$ are all monic and have the same degree.

Theorems & Definitions (106)

  • Definition 1
  • Proposition 2.1
  • Definition 2
  • Definition 3
  • Proposition 2.2
  • proof
  • Definition 4: $\text{Factorization}^{\prime}$ step
  • Remark 2.3
  • Definition 5: Factorization Step
  • Remark 2.4
  • ...and 96 more