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This question is motivated by recent discovery of Andrew Booker+Andrew Sutherland.

Richard Guy's problem D5 in his Unsolved Problems in Number Theory contains the original question for the sum of three cubes and a few other interesting problems, e.g. the solutions of Diophantine equation $$x^3+y^3+2z^3=k,$$ where $k$ is an integer neither a cube nor twice a cube. For $k<1000$, only two candidates $k=148, 671$ are left(see Tomita Seiji's database).

Question: Is there any known computational effort for these two Diophantine equations?

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Above equation shown below:

           $x^3+y^3+2z^3=K$ -----$(1)$

Allan Macleod has given a method for finding

numerical solutions to equation $(1)$.

The link to his article is given below.

    https://arxiv.org/pdf/1109.2396.pdf

If "OP" has done some research on equation $(1)$ then

maybe he can post it on "math overflow" website.

That way it may help others to find more numerical

solutions to the unknown values of $'K'$ in equation $(1)$.

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