We try to find positive integer solutions (or decide if there are any) for
$$ \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = n $$
for any given positive integer n.
In the first answer at https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4 Alon Amit seems to claim that this problem is uncomputable, without proof or reasoning.
This is a striking example of the way diophantine equations with tiny coefficients can have enormous solutions. This isn’t merely awe-inspiring, it is profound. The negative solution of Hilbert’s 10th problem means that the growth of the solutions as the coefficients get larger is an uncomputable function, for if it were computable, we would have had a simple algorithm for solving diophantine equations, and there isn’t one (simple or complex). Here, the correspondence 4→ 80-digit numbers, 178→ hundreds-of-millions-digit numbers and 896→ trillions of digits gives us a glimpse into the first few tiny steps of that monstrous, uncomputable function. Tweak the numbers in your equation, and the solutions promptly exceed anything that fits in our puny little universe. What a wonderfully sneaky little equation.
If it's indeed the case, that would indeed explain why these solutions are so big (not necessary for n=4, but from a given n).
I wonder, if it's true?