## 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.