1. ## Classic Diophantine problem

For two postage stamp values, one of *n* cents and one of *m* cents
the following Diophantine equation

mx + ny = T

gives the total value of postage or T, where x and y represent the number of each denomination of stamps
and are therefore integers. Clearly, you could get many values for T.

But there are also many values of T that you could NOT get.

I have two questions regarding this situation:

1) Under what conditions will stamp amounts *m* and *n* generate all but a finite number of
postage stamp amounts?

2) Under the conditions of question one (the stamps generate all but a finite number of amounts)
what is a *formula* for the largest postage value that CANNOT be generated?

Any help??

• If $\displaystyle (m,n)=M>1$ then $\displaystyle M$ divides any linear combination (with natural coefficients) of them, so we do not generate the numbers which are not multiples of M and so there are infinitely many numbers left out.
• If $\displaystyle (m,n)=1$, read here