Given relatively prime positive integers a₁ , …, a_n , we call an integer t representable if there exist nonnegative integers m₁ , …, m_n such that

t = m₁ a₁ + … + m_n a_n .
We study the linear diophantine problem of Frobenius: namely, to find the largest integer which is not representable.

We translate this problem into a geometric one: consider N(t), the number of nonnegative integer solutions (m₁ , …, m_n ) to m₁ a₁ + … + m_n a_n = t for any positive integer t. N(t) enumerates the integer points in a polytope. Solving the Frobenius problem now simply means finding the largest zero of N(t). N(t) turns out to be a quasi-polynomial, thereby yielding a straightforward analytic tool to recover and extend some well-known results on this classical problem.