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.
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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.