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I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that $x^3+y^3+z^3=c$. If the status is unknown, what are the conjectures or consensuses on the decidabiity of such small cubic equations?

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    $\begingroup$ Poonen's article "Undecidability in number theory" begins: "Does the equation $x^3+y^3+z^3 = 29$ have a solution in integers? Yes: $(3, 1, 1)$, for instance. How about $x^3+y^3+z^3 = 30$? Again yes, although this was not known until 1999: the smallest solution is $(−283059965, −2218888517, 2220422932)$. And how about $x^3+y^3+z^3 = 33$? This is an unsolved problem." $\endgroup$
    – Steve Huntsman
    Commented Apr 26, 2015 at 16:29
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    $\begingroup$ The latest relevant reference seems to be Elsenhans, A.-S. and Jahnel, J. "New sums of three cubes". Math. Comp. 78, 1227 (2009). $\endgroup$
    – Steve Huntsman
    Commented Apr 26, 2015 at 16:36
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    $\begingroup$ See also mathoverflow.net/questions/138886/… . To my knowledge, it is expected that the only obstruction is the obvious 3-adic condition that $c \not\equiv 4,5 \bmod 9$, but a proof seems to be far away. If this expectation is true, then of course there is a pretty trivial Turing machine that will decide the question... Colliot-Thélène and Wittenberg [Groupe de Brauer et points entiers..., Amer. J. Math. 134 (2012), no. 5, 1303–1327] have shown that there is no Brauer-Manin obstruction for the other $c$. $\endgroup$
    – Michael Stoll
    Commented Apr 26, 2015 at 17:51

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