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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, namely $$(154476802108746166441951315019919837485664325669565431700026634898253202035277999,$$ $$36875131794129999827197811565225474825492979968971970996283137471637224634055579 ,$$ $$ 4373612677928697257861252602371390152816537558161613618621437993378423467772036) .$$ This leads to the following general question: Is the set $C\subseteq {\mathbb N}$ defined by $$ C = \{n\in\mathbb{N}\setminus\{0\}: (\exists a,b,c \in\mathbb{N}\setminus\{0\}):\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = n\}$$ computable? (As user Watson points out in the comment section below, $C$ contains no odd numbers. It would also be great to see an even number $\geq 6$ not contained in $C$.)

YCor
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Dominic van der Zypen
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  • Oops thanks - will remove that second question! – Dominic van der Zypen Aug 14 '17 at 19:19
  • A quick heuristic comment: for each $n$, the existence of such $a, b, c$ corresponds to the existence of a solution to a degree-$3$ Diophantine equation in $3$ variables; and this is, I believe, a bit beyond what is generally known to be decidable. So I suspect that there will be no general reason why this set is computable; rather, if it is computable (which I extremely strongly suspect it is), the proof is likely to be a corollary of a complete characterization of these $n$s, which will not use any computability theory. – Noah Schweber Aug 14 '17 at 19:23
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    I don't have the time to look into the details, but from a cursory glance at this quora answer it looks like the problem amounts to deciding whether a certain elliptic curve has solutions over $\mathbb{Q}$. Now IIRC there is an algorithm for that (find solutions locally and try to globalize) provided Ш is finite, which is conjecturally always the case. So I think conjecturally your set is indeed computable. – Gro-Tsen Aug 14 '17 at 19:28
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    See also https://math.stackexchange.com/questions/402537/find-integer-in-the-form-fracabc-fracbca-fraccab/409450 and Bremner and Macleod, An unusual cubic representation problem, Annales Mathematicae et Informaticae 43 (2014) 29-41, http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf – Gerry Myerson Aug 15 '17 at 05:43
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    Here is explained why $C$ doesn't contain any odd number. – Watson Aug 19 '17 at 20:30
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    So we're half done! – Joel David Hamkins Sep 14 '17 at 12:28
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    This is somewhat similar to the question which natural numbers $n$ are congruent numbers (i.e. the area of a right-angled triangle with rational sides), which comes down to asking whether the elliptic curve $E_n \colon y^2 = x^3-n^2x$ has positive rank over $\mathbb Q$. In this case, there is a conjectural answer (Tunnell's Theorem, conditional on the BSD conjecture), which relies on the fact that the $E_n$ are all quadratic twists of a fixed curve. The question asked here is likely to be harder, since the resulting curves are not twists, and there is the positivity condition. --> – Michael Stoll Oct 26 '17 at 15:29
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    (continued) Note that there are odd $n$ for which the associated elliptic curve has positive rank, but does not have rational points on the component that contains the positive points. (IIRC, $n = 19$ is an example.) This gives an additional twist to the question. – Michael Stoll Oct 26 '17 at 15:31
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    Cross-posted to https://cstheory.stackexchange.com/questions/39383/computability-of-a-weird-set – Emil Jeřábek Oct 26 '17 at 16:05
  • Also discussed at https://mathematica.stackexchange.com/questions/184956/apple-banana-pineapple-problem – Gerry Myerson Sep 16 '20 at 11:18
  • https://mathoverflow.net/questions/264754/solution-to-a-diophantine-equation/275193#275193 – individ Sep 18 '20 at 04:14

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