A Study on the Estimation of the Frobenius Numbers Generated by Binomial Coefficients Using Linear Regression

The Frobenius problem is a classical problem in number theory and combinatorics that explores the range and maximum values of integers that can be represented as combinations of a given set of integers. There is a simple formula for the Frobenius Number for the case of two integers. There are just some special results for the cases of three or more integers, and the general formula has not been discovered. In this paper, we study the approximation of Asymptotic behavior using Linear Regression to get a Frobenius Number for one existing and a new result. Initially, finding the Frobenius number required a lot of computation, including finding an Apery set. Still, we took advantage of the fact that the Frobenius number can be found directly by making a function prediction using the individual data of the found Frobenius numbers. The main reason why function prediction in this way can be correct is that the calculation to find the Frobenius number involves a non-negative integer combination of the elements of the numerical semigroup, so if we think of it as a non-negative integer combination with the coefficients of the integers closest to the function found in Linear Regression, we can get a predicted function that is expected to be accurate. The methodology of this study may not be well applied to functions with general real numbers. Still, we found that if we analyze discrete values well, we can get a sufficiently predicted function.