We have now found all solutions to the Diophantine equation. For each \(k\text{,}\) \(x = -1-29k\) and \(y = 2 + 17k\) will satisfy the equation. We could check this for a few cases. If \(k = 0\text{,}\) the solution is \((-1,2)\text{,}\) and yes, \(-17 + 2\cdot 29 = 41\text{.}\) If \(k = 3\text{,}\) the solution is \((-88, 53)\text{.}\) If \(k = -2\text{,}\) we get \((57, -32)\text{.}\)
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