Title

Cauchy-Mirimanoff Polynomials

Major

Applied Mathematics

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Dr. Kelly Pearson

Presentation Format

Event

Abstract/Description

For an integer n > 2 define Pn (X) = (X + 1)n – Xn – 1. Let En (X) be the remaining factor of Pn (X) in Q [X] after removing X and the cyclotomic factors X + 1 and X2 + X + 1. Then Pn (X) = X(X+1)εn (X2 + X + 1)δn En (X) where for even n εn = δn = 0; for odd n εn = 1 and δn = 0,1,2 according as n = 0, 2, 1 (mod 3). In 1903 Mirimanoff conjectured the irreducibility of En (X) over Q when n is prime. This talk will focus on eliminating any factors of degree six. A characterization of the only possible factors of Pn that are of degree six will be given as well as the primes for which these polynomials are possible factors of En.

Other Affiliations

Science and Mathematics

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Cauchy-Mirimanoff Polynomials

For an integer n > 2 define Pn (X) = (X + 1)n – Xn – 1. Let En (X) be the remaining factor of Pn (X) in Q [X] after removing X and the cyclotomic factors X + 1 and X2 + X + 1. Then Pn (X) = X(X+1)εn (X2 + X + 1)δn En (X) where for even n εn = δn = 0; for odd n εn = 1 and δn = 0,1,2 according as n = 0, 2, 1 (mod 3). In 1903 Mirimanoff conjectured the irreducibility of En (X) over Q when n is prime. This talk will focus on eliminating any factors of degree six. A characterization of the only possible factors of Pn that are of degree six will be given as well as the primes for which these polynomials are possible factors of En.