Abstract:  Here is a 19^th century theorem due to Sylvester, and to Frobenius, and probably to others:  If A is a finite set of non-negative integers with greatest common divisor 1, then every sufficiently large positive integer is representable as a linear combination of the elements of A with non-negative integer coefficients.  In this case, the Frobenius problem associated with A is to find the largest integer not so representable. We will see that there is a way to reframe that 19^th century theorem so that analogous theorems hold in some rings other than the integers, and analogous problems can be posed in those rings.

 

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