Help solving this Diophantine equation

For a problem that I'm working on, I need to solve this Diophantine equation:-

$ -2a^3 + b^3 + c^3 = 36650$, where $a, b, c > 0$ are all DISTINCT positive integers, and $a, b, c \notin$ { 2, 9, 15, 16, 33, 34}

How does one go about solving this? Is brute-force the only possible way? Or could there be a case that no integer solutions exist for this equation?

Also, are there any online computing engines, that allow me to set constraints, and solve Diophantine equations of this sort?

Any and all help is appreciated! Thanks!

Answer

For your modified question $a^3 + b^3 + c^3 - 3d = -83449$, there are in fact infinite solutions.

Considered modulo 3,

$$a^3 + b^3 + c^3 \equiv 2 \mod 3$$

Now, we know that modulo 3, any cube is congruent to itself, or $x^3 \equiv x \mod 3$.

Therefore, the above equation reduces to

$$a+b+c \equiv 2 \mod 3$$

One triplet of $(a,b,c)$ that satisfies the above equation is $(1,3,4)$. We can then calculate the value of $d$, which is simply $27847$ in this case.

Rearranging the above equation, since $3d=a^3+b^3+c^3+83449$, $d$ is positive for any positive values of $a,b,c$, allowing us to conclude that there are infinite solutions.