Numerical Semigroups

You're craving this, and only this: exactly 23 chicken nuggets. However the local Mexican restaurant, M.C. Donaldo's, only sells chicken nuggets in quantities of either 4 or 9. Is there any way the restaurant can sell you the right combination that will total up to 23? Unfortunately, you're out of luck. It turns out if you had wanted any amount of chicken nuggets larger than 23, then your favorite bistro could've helped you out. This chicken nugget problem of yours is sometimes called the Frobenius problem or coin problem: Given a set of relatively prime integers in the natural numbers, what is the smallest number not in the semigroup generated by these integers? The latter name for this problem comes from a problem involving currency, where given various coins of different denominations, what's the largest monetary value that can't be obtained using these coins?

For this project, we'll investigate the Frobenius problem and look at some of the progress that has been made on it so far. Knowing a little bit of number theory can help, but is not necessary.