  For given coprime positive integers $a$ and $b$, the classical Frobenius coin
problem asked to find the largest number that cannot be expressed in the form
$ax+by$ for nonnegative integers $x$ and $y$, also known as the Frobenius
number. Sylvester answered this question and also discovered the number of
these nonrepresentable numbers. In recent times, a lot of progress has been
made regarding the sums of a fixed power of these nonrepresentable numbers,
commonly known as the Sylvester sums. However, the actual list of
nonrepresentable numbers has remained mysterious so far. In this note, we
obtain a complete list, that is a simple explicit description of these
nonrepresentable numbers. We give two different proofs of our result. The first
one enables us to use Eisenstein's lattice point counting techniques to study
the set of nonrepresentable numbers, whereas the second one is a short direct
proof.
