Find all integer solutions to Diophantine equation $x^3+y^3+z^3=w^3$

Question

Compute all integer solutions to the equation

$$x^3+y^3+z^3=w^3$$

Answer 1

(I revisited and revised this old question because of this post.)

The complete rational solution to, $$x_1^3+x_2^3+x_3^3+x_4^3=0\tag1$$

is known and there are various formulas (due to Euler, Binet, Steggall, Elkies, etc).

I. Integer case:

However, if we wish to solve the form,

$$x_1^3+x_2^3+x_3^3 = x_4^3\tag2$$

where $0<x_1<x_2<x_3$, then there is also a complete positive integer solution given by Choudhry's On Equal Sums of Cubes (1998). For those who want the summarized version, for positive integers $a,b,c$,

$$\begin{aligned} d\,x_1 &= (-a^3 - b^3 + c^3)c\\ d\,x_2 &= -(a^2 - a b + b^2)^2 + (a + b)c^3\\ d\,x_3 &= (a^2 - a b + b^2)^2 + (2a - b)c^3\\ d\,x_4 &= (a^3 + (a - b)^3 + c^3)c\end{aligned}$$

and,

$$a>b,\quad c >(a^3+b^3)^{1/3}$$

where $d=1$, or $d$ is chosen such that $\text{GCD}(x_1,x_2,x_3,x_4)=1$.

II. Example:

Given any solution $x_1,x_2,x_3,x_4$, then one can recover $a,b$ as,

$$(x_2 + x_3) (a x_2 - b x_3)^3 + (a^2 - a b + b^2)\, x_1^3\, \big((2 a - b) x_2 - (a + b) x_3\big)=0\tag3$$

then $c,d$,

$$3 a c^2 x_1 + (a^3 + b^3 - c^3) (x_2 + x_3)=0\tag4$$

$$3 a (a^2 - a b + b^2)^2 + d \, \big((2 a - b) x_2 - (a + b) x_3\big)=0\tag5$$

For example, the smallest is the well-known,

$$3^3+4^3+5^3 = 6^3$$

Substituting $x_1,\,x_2,\,x_3 = 3,4,5$ into $(3)$, one finds the linear factor,

$$(a-2b)=0$$

so $a=2,\;b=1$. Then into $(4)$ and $(5)$,

$$(c-3)=0$$

$$(d-18) = 0$$

III. Statistics:

A lot of positive integers $N^3$ can be expressed as a sum of three positive cubes. The sequence A023042 begins as,

$$N= 6, 9, 12, 18, 19, 20, 24, 25, 27, 28, 29, 30, 36, 38, 40,\dots$$

and for a higher range,

$$N=\dots,9990, 9991, 9992, 9993, 9994, 9995, 9996, 9997, 9998, 9999, 10000,\dots$$

which shows no skip. In fact, based on the table here, it is $94.2\text{%}$ of all $N<10000$. Extrapolating from the trends in the table, it is easily $99\text{%}$ of all $N<1000000$. Thus, choosing a random $N$ in the high end of the range, chances are very good there will be a positive integer solution to,

$$x_1^3+x_2^3+x_3^3 = N^3$$

P.S. Thanks, ccorn!