Anika O'Donnell: Generalized Frobenius numbers: asymptotics and product families

Given d positive integers a₁,a₂,...,a_d such that gcd(a₁,a₂,...,a_d)=1, the Frobenius coin-exchange problem asks to find the largest number n that does not have a nonnegative integer solution (x₁,x₂,...,x_d) to the equation n=a₁x₁+a₂x₂+...+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)!a₁a₂...a_d)^1/(d-1). We also find explicit bounds for the generalized Frobenius number in three specific cases.