Coin Problem

Let there be $n\geq 2$ Integers $0<a_1<\ldots<a_n$ with $(a_1,a_2,\ldots,a_n)=1$ (all Relatively Prime). For large enough $N=\sum_{i=1}^n a_ix_i$, there is a solution in Nonnegative Integers $x_i$. The greatest $N=g(a_1, a_2, \dots a_n)$ for which there is no solution is called the coin problem. Sylvester showed

$$g(a_1,a_2)=(a_1-1)(a_2-1)-1,$$

and an explicit solution is known for $n=3$, but no closed form solution is known for larger $N$.

References

Guy, R. K. ``The Money-Changing Problem.'' §C7 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 113-114, 1994.