SUM OF THREE CUBES

[Harvey Friedman]

https://www.ams.org/journals/tran/2005-357-03/S0002-9947-04-03632-3/S0002-9947-04-03632-3.pdfhttps://phys.org/news/2019-09-sum-cubes-solvedusing-real-life.html
http://nautil.us/issue/70/variables/how-search-algorithms-are-changing-the-course-of-mathematics
https://math.mit.edu/~drew/Waterloo2019.pdf

It has been determined which integers from 0 through 100 are or are
not the sum of 3 cubes of integers.

For 33:

X = 8,866,128,975,287,528
Y = -8,778,405,442,862,239
Z = –2,736,111,468,807,040

For 42:

X = -80538738812075974
Y = 80435758145817515
Z = 12602123297335631

QUESTION: Is the set of all sums of three cubes of integers recursive?
It is obviously recursively enumerable.

This seems to be the most attractive set of integers for which it is
not known whether it is recursive. Any competitors?

QUESTION: Is there an integer such that the question of whether it is
the sum of three cubes is independent of ZFC?

If the second question has no for its answer, then the first question
has yes for its answer.

I got interested in the question of giving natural sets of integers
KNOWN to be nonrecursive. See
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.170.8659&rep=rep1&type=pdf
https://www.ams.org/journals/tran/2005-357-03/S0002-9947-04-03632-3/S0002-9947-04-03632-3.pdf

Harvey Friedman