Gauss-Legendre's theorem on sums of three squares states that

$$ {x ^ 2 + y ^ 2 + z ^ 2: x, y, z in mathbb Z } = mathbb N setminus {4 ^ k (8m + 7): k, m in mathbb N }, $$ from where $ mathbb N = {0,1,2, ldots } $,

It's easy to see that set $ {x ^ 3 + y ^ 3 + z ^ 3: x, y, z in mathbb Z } $ does not contain an integer that is congruent $ 4 $ or $ -4 $ modulo $ 9 $, In 1992, Heath-Brown suspected that every integer $ m not equiv pm4 pmod9 $ can be written as $ x ^ 3 + y ^ 3 + z ^ 3 $ With $ x, y, z in mathbb Z $, Recently, A.R. Booker arXiv: 1903.04284 integers $ x, y, z $ With $ x ^ 3 + y ^ 3 + z ^ 3 = $ 33,

It is well known that

$$ left { binom x2 + binom y2 + binom z2: x, y, z in mathbb Z right } = mathbb N, $$ which was claimed by Fermat and proved by Gauss.

Here I ask a similar question.

**question**: Has the set $ { binom x3 + binom y3 + binom z3: x, y, z in mathbb Z } $ contain all integers?

Clear,

$ binom {-x} 3 = – binom {x + 2} 3. $ About Mathematica I found that the only integers under $ 0, ldots, $ 2000 not in the set

$$ left { binom x3 + binom y3 + binom z3: x, y, z in {- 600, ldots, 600 } right } $$

are

$ 522, , 523, , 622, , 633, 642, 843, 863, 918, , 1013, , 1458, 1523, , 1878. $$

For example,

$$ 183 = binom {549} 3+ binom {-525} 3+ binom {-266} 3 $$

and

$$ 423 = binom {426} 3+ binom {-416} 3+ binom {-161} 3. $$

In my opinion, the question has a positive answer. Your comments are welcome!