# Linear diophantine equation in n variables

Let n>3. Is there any way to generate all integer solutions of linear diophantine equation in n variables, or at least to determine number of such solutions?

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The number of solutions is either zero or infinite. As for the ways to generate them, yes there are many ways. Look at Morris Newman's "Integer matrices", and check out "Hermite Normal Form".

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Say if $\sum_{i=1}^k a_ix_i = N$ is the equation with $gcd(a_i,a_j)=1$ if $i\neq j$, then if we seek solutions $x_i:0\leq x_i<a_i$, then is the solution unique? –  Turbo Sep 14 '13 at 17:15