Diophantine equations whose coefficients are increasing sequences of integers

Question

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.

Answer 1

Linear Diophantine equations are always solvable decidable (in linear time). If the coefficients are $a_1, a_2, ... a_n$ then the numbers that can appear on the RHS are precisely the multiples of $\text{gcd}(a_1, ... a_n)$, and one can find solutions using the extended Euclidean algorithm.