Generalized Frobenius Numbers: Asymptotics and Two Product Families
Author: Anika O'Donnell
Faculty Supervisor: Matthias Beck
Department: Mathematics
Given d positive integers a_1, a_2, ..., a_d such that gcd(a_1, a_2, ..., a_d) = 1, the Frobenius coin-exchange problem asks to find the largest number n that does not have a nonnegative integer solution (x_1, x_2, ..., x_d) to the equation n = a_1x_1 + a_2x_2 + ... + a_dx_d. The generalized Frobenius problem asks to find the largest number n that does not have more than s distinct solutions to the above equation. We prove that the generalized Frobenius number grows asymptotically like (s(d-1)!a_1a_2...a_n)^(1/(d-1)). We also find explicit bounds for the generalized Frobenius number in three specific cases.