A simple polynomial for a simple transposition

2006/03/02 by Greg Martin, Martin, Greg
Computer Science · Mathematics · #11A07 #11T06 #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11A07 #msc:11T06

paper · pdf · doi:10.48550/arxiv.math/0603054

4 pages

arxiv created 2006/03/02 · openalex publication_date 2006/03/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we review some facts about polynomials representing functions modulo primes p. In addition we prove that the polynomial f(x) = x^p-2 + x^p-3 + ... + x³ + x² + 2x + 1 represents the transposition (0 1) modulo p, that is, f(0) ≡ 1 (mod p), f(1) ≡ 0 (mod p), and f(a) ≡ a (mod p) for all 2 ≤ a ≤ p-1.

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