Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Euler's totient function of polynomials over finite fields

Xiumei Li, Min Sha

TL;DR

The paper develops a comprehensive study of Euler's totient function for polynomials over finite fields, denoted $\Phi(f)$, and analyzes polynomial analogues of classical totient conjectures. It derives a concrete multiplicative formula for $\Phi(f)$, analyzes collisions of totient values via irreducible-divisor data, and characterizes when two polynomials share the same totient value across different $q$-regimes. It then establishes polynomial versions of Carmichael's, Sierpiński's, and Erdős' conjectures, including uniqueness criteria for preimages, realizability of prescribed preimage sizes, and explicit intersections with the sum-of-divisors values, along with a rigorous distribution result showing the density of $\Phi(\mathbb{A})$ is zero. Collectively, the results illuminate the arithmetic of $\Phi$ on $\mathbb{F}_q[x]$, reveal intricate dependence on $q$, and demonstrate the sparsity of totient values in the polynomial setting.

Abstract

In this paper, we study some typical arithmetic properties of Euler's totient function of polynomials over finite fields. Especially, we study polynomial analogues of some classical conjectures about Euler's totient function, such as Carmichael's conjecture, Sierpiński's conjecture, and Erdös' conjecture.

On Euler's totient function of polynomials over finite fields

TL;DR

The paper develops a comprehensive study of Euler's totient function for polynomials over finite fields, denoted , and analyzes polynomial analogues of classical totient conjectures. It derives a concrete multiplicative formula for , analyzes collisions of totient values via irreducible-divisor data, and characterizes when two polynomials share the same totient value across different -regimes. It then establishes polynomial versions of Carmichael's, Sierpiński's, and Erdős' conjectures, including uniqueness criteria for preimages, realizability of prescribed preimage sizes, and explicit intersections with the sum-of-divisors values, along with a rigorous distribution result showing the density of is zero. Collectively, the results illuminate the arithmetic of on , reveal intricate dependence on , and demonstrate the sparsity of totient values in the polynomial setting.

Abstract

In this paper, we study some typical arithmetic properties of Euler's totient function of polynomials over finite fields. Especially, we study polynomial analogues of some classical conjectures about Euler's totient function, such as Carmichael's conjecture, Sierpiński's conjecture, and Erdös' conjecture.
Paper Structure (7 sections, 21 theorems, 72 equations)

This paper contains 7 sections, 21 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Collision of totient values
  4. Carmichael's conjecture
  5. Sierpiński's conjecture
  6. Erdös' conjecture
  7. Distribution of the values of $\Phi$

Key Result

Lemma 2.1

Let $a,b\in \mathbb{N}$ such that $\gcd(a,b)=1$ and $n\in \mathbb{N}, n >1$. Then, there exists a prime divisor of $a^n-b^n$ that does not divide $a^k-b^k$ for all $k \in \{1,2,\cdots,n-1\}$$($we call it a primitive prime divisor$)$, except exactly in the following cases:

Theorems & Definitions (36)

  • Lemma 2.1: Zsigmondy's theorem
  • Lemma 2.2: Stirling's formula
  • Lemma 2.3: BD
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • proof
  • Proposition 4.1
  • ...and 26 more