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Let A(n) be a $k\times s$ matrix and $m(n)$ be a $k$ dimensional vector,where all entries of A(n) and $m(n)$ are integer-valued polynomials in $n$.Suppose that $$t(m(n)|A(n))=#\{x\in\mathbb{Z}_{+}^{s}\mid A(n)x=m(n)\}$$ isfinite for each $n\in \mathbb{N}$, where $Z_+$ is the set of nonnegativeintegers. This paper conjectures that $t(m(n)|A(n))$ is an integer-valuedquasi-polynomial in $n$ for $n$ sufficiently large and verifies the conjecturein several cases.

On a Conjecture about the Number of Solutions to Linear Diophantine Equations with a Positive Integer Parameter