I needed (for my research) to solve a Diophantine equation, in particular, $$ 2 a + 3 b + 4 c + 5 d = 12 .$$ And I could easily solve it (for example, on solution is $a=2, b=1, c=0, d=1$). But this made me wonder if such equations, with their coefficients increasing sequences of natural numbers, are a special case of Diophantine equations that are always explicitly solvable, despite the negative solution to Hilbert's 10th problem.
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Linear Diophantine equations are always