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I'm looking to see if there is an integral solution to $f(x,y,z)=n$ where f is a cubic form. Especially interesting is the diagonal case:

$$ ax^3+by^3+cz^3=n $$

for fixed integers $a,b,c,n$. If there are no modular obstacles -- if $f(x,y,z)\equiv n \pmod m$ for all integers $m>0$ -- then what techniques are available? I'm only interested in learning if there are solutions; I don't need to enumerate solutions. I think that for very many cases there are no small solutions but solutions still exist, since you can search for ever-larger $|x|,|y|,|z|$.

Charles
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    That the equation $x^3 + y^3 + z^3 = 42$ has a non-trivial integral solution (previously it was not known if any solutions existed but it is widely expected to exist) is considered one of the biggest mathematical breakthroughs of 2019, and it took 65 years to find a non-trivial solution to the equation $x^3 + y^3 + z^3 = 3$. I don't think there's an easy answer to your question. – Stanley Yao Xiao Mar 30 '22 at 20:48
  • References for above results can be found in https://en.wikipedia.org/wiki/Sums_of_three_cubes – Max Alekseyev Mar 30 '22 at 21:08
  • @StanleyYaoXiao Definitely no easy answers here -- this is in the no-man's land beyond elliptic curves. $x^3+y^2+z^3=n$ has been attacked, to my knowledge, with slightly sophisticated brute force searches. Discussion of the techniques used there would certainly be in-bounds, as would applying techniques like Broughan's on the sum of two cubes. I'm just generally getting an idea for what's out there for problems of this type. – Charles Mar 30 '22 at 21:56
  • I think the sum-of-three-cubes question has been tackled with searches that are more than just slightly sophisticated. Do look at the references on that problem. Maybe you'll find a way to apply the methods to more general (diagonal) ternary cubics. – Gerry Myerson Mar 31 '22 at 02:34
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    @GerryMyerson I've been looking and you're right, there's a lot more going on in those searches than I had remembered. – Charles Mar 31 '22 at 02:47
  • You may find Selmer's 160 page "The Diophantine Equation ax^3+by^3+cz^3=0" Acta Math. (1951) article of interest. – Somos Apr 01 '22 at 01:31
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    Until recently, the smallest open equation of this type has been $x^3+y^3+z^3+xyz=5$. This is now solved, see https://mathoverflow.net/questions/400714/can-you-solve-the-listed-smallest-open-diophantine-equations . The trick which I used to facilitate the computer search for solution of this equation can be used to solve many other equations as well, so please have a look. – Bogdan Grechuk Apr 01 '22 at 12:17

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