Number of non-negative integer solutions for linear equations with constants

Question

How do we find the number of non-negative integer solutions for linear equation of the form:

$$a \cdot x + b \cdot y = c$$

Where $a, b, c$ are constants and $x,y$ are the variables ?

Answer 1

Not a complete answer, but a relatively simple one and approximate one. By Schur's theorem of combinatorics^?, the number of solutions is asymptotically ($c \to \infty$):

$$ \frac{c}{ab} $$

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Schur's theorem of combinatorics states that the number of solutions of (with $a_i$ relatively prime):

$$ \sum_{i=1}^M a_i x_i = c $$

is:

$$ \frac{c^{M-1}}{(M-1)!\prod a_i} $$

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^? This name is used by Wilf's Generatingfunctionology, but I cannot seem to find it elsewhere. It appears that Schur has many theorems.

Answer 2

You might be looking for Diophantine equations.

Check this link or this for an explanation!