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.