A, B, C, D are all distinct digits. ABCD is a concatenated number
$ABCD$ = $A^3$ + $(A+C+D)^3 $ = $D^3+ (A + D)^3$
Figure out this Number and why is it Famous?
Also provide detailed reasoning to show how you figured it out.
The answer is
Which is the famous Taxicab number. It is the smallest integer that can be expressed as a sum of two positive integer cubes in 2 different ways.
ABCD is 1729, the sum of the cubes of 1 and 1+2+9=12 and also of the cubes of 9 and 1+9=10. It's famous because of the story about Hardy, Ramanujan and the taxi. I think it's rather unfair to call it a "cube" since it isn't itself a cube. No computers required precisely because the story is so famous and it was the first thing I thought of on looking at the puzzle.
here's a proof that it's the only solution. From the second equation we know that $1000A\leq 9^3+(A+9)^3$, which requires $A=1$ or $A=2$; if $A=2$ then our number has to be $9^3+11^3=2060$, which plainly doesn't work out. So $A=1$ and we have $1BCD=(C+D+1)^3+1=(D+1)^3+D^3$. Now $D$ has to be at least 8 to make that last thing have four digits, so we just have to check 8 (which doesn't work) and 9 (which does).
I have to say, though, that
precisely because this is so well known and the question didn't ask for a uniqueness proof or anything of the sort, you should accept Kruga's answer which delivered everything you asked for and I'm sure wasn't arrived at by computerized calculation.