A CLASS OF IRREDUCIBLE POLYNOMIALS ASSOCIATED WITH PRIME DIVISORS OF VALUES OF CYCLOTOMIC POLYNOMIALS

We prove that for every sufficiently large integer n , the polynomial 1 + x + x 2 / 11 + x 3 / 111 + ⋯ + x n / 111 … 1 is irreducible over the rationals, where the coefficient of x k for 1 ⩽ k ⩽ n is the reciprocal of the decimal number consisting of k digits which are each 1 . Similar results following from the same techniques are discussed.