A ug 2 00 7 A property of cyclotomic polynomials 1

Given two cyclotomic polynomials Φn(x) and Φm(x), n 6= m, we determine the minimal natural number k such that we can write k = a(x)Φn(x) + b(x)Φm(x), with a(x) and b(x) integer polynomials. In June 2000 the author of this note gave a talk at the conference ”Combinatorics 2000” in Gaeta (Italy), titled Dividing Cyclotomic Polynomials. The content of that talk was proposed to the American Mathematical Monthly as Problem 10914 in Volume 109, January 2002, and a composite solution by R. Stong and N. Komanda was published on Volume 110, October 2003. The problem was also solved by R. Chapman, C. P. Rupert, the GCHQ Problem Solving Group and the proposer, whose original solution is given in this note. A property of cyclotomic polynomials. We denote, as usual, with Φk(x) the cyclotomic polynomial (with integer coefficients) of index k, defined inductively through the identity x − 1 = ∏ Φk(x), where k runs among the divisors of m. The basic properties of cyclotomic polynomials are well known, as well as the role they play in several branches of Mathematics. Here we just recall that Φm(1) = p, if m is a p-power (p prime), Φm(1) = 1 otherwise. Given two different cyclotomic polynomials Φm(x),Φn(x), we can find two polynomials s(x), t(x) ∈ Q[x] such that 1 = s(x)Φm(x) + t(x)Φn(x), Φm(x) and Φn(x) being irreducible as element of the euclidean ring Q[x]. Since the ring Z[x] is a unique factorization domain, but not a euclidean 1AMS MCS 11C08. 2Supported by M.I.U.R., Università di Palermo.