(a+b)$ computable?

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Sep 18, 2020 at 4:14	comment	added	individ		mathoverflow.net/questions/264754/…
Sep 16, 2020 at 11:18	comment	added	Gerry Myerson		Also discussed at mathematica.stackexchange.com/questions/184956/…
Sep 16, 2020 at 8:27	history	edited	YCor	CC BY-SA 4.0	added values, made title more specific
S Sep 16, 2020 at 8:16	history	edited	Glorfindel	CC BY-SA 4.0	broken link fixed
S Sep 16, 2020 at 8:16	history	suggested	JimN	CC BY-SA 4.0	Replaced a broken link with a different source that contains the absurdly high smallest positive integer solution
Sep 16, 2020 at 5:12	review	Suggested edits
S Sep 16, 2020 at 8:16
Nov 16, 2017 at 15:47	history	edited	Dominic van der Zypen	CC BY-SA 3.0	Included Watson's comment
Oct 26, 2017 at 16:05	comment	added	Emil Jeřábek		Cross-posted to cstheory.stackexchange.com/questions/39383/…
Oct 26, 2017 at 15:31	comment	added	Michael Stoll		(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.
Oct 26, 2017 at 15:29	comment	added	Michael Stoll		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. -->
Oct 26, 2017 at 14:34	history	edited	Dominic van der Zypen	CC BY-SA 3.0	added 1 character in body
Sep 14, 2017 at 12:28	comment	added	Joel David Hamkins		So we're half done!
Sep 14, 2017 at 6:53	history	edited	Dominic van der Zypen	CC BY-SA 3.0	added 36 characters in body
Aug 19, 2017 at 20:30	comment	added	Watson		Here is explained why $C$ doesn't contain any odd number.
Aug 15, 2017 at 5:43	comment	added	Gerry Myerson		See also math.stackexchange.com/questions/402537/… and Bremner and Macleod, An unusual cubic representation problem, Annales Mathematicae et Informaticae 43 (2014) 29-41, ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf
Aug 14, 2017 at 19:28	comment	added	Gro-Tsen		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.
Aug 14, 2017 at 19:23	comment	added	Noah Schweber		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.
Aug 14, 2017 at 19:20	history	edited	Dominic van der Zypen	CC BY-SA 3.0	deleted 77 characters in body
Aug 14, 2017 at 19:19	comment	added	Dominic van der Zypen		Oops thanks - will remove that second question!
Aug 14, 2017 at 19:16	history	asked	Dominic van der Zypen	CC BY-SA 3.0

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