# Computability of a "weird" set

The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying

$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$

are absurdly high. This leads to the following general question: Is the set $C\subseteq {\mathbb N}$ defined by $$C = \left\{n\in\mathbb{N}\setminus\{0\}: \big(\exists a,b,c \in\mathbb{N}\setminus\{0\}\big):\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = n\right\}$$ computable?

• I'm not an expert, but if you simplify the equation you get a cubic diophantine equation in three variables, and I think it's an open problem if universality can be achieved in such a setting. In every case, quadratic diophantine equations in two variables ($ax^2 + by + c = 0$) are already NP-complete (see NP-complete decision problems for quadratic polynomials) ... so I'm not surprised if switching to cubic+three vars leads to very hard instances (outside NP). Commented Oct 26, 2017 at 15:39
• Cross-posted from mathoverflow.net/questions/278747/is-this-set-computable Commented Oct 26, 2017 at 16:05