Unconventional Hardy-Ramanujan number $a^3 + b^3 = c^3 + c^3$ - can it's existence / non-existence be proven? 
Can it be proven that a number exists such that 
  $$\text{number} = a^3 + b^3 = c^3 + c^3,$$ where $a,b$ and $c$ are $3$ distinct positive integers?

If it cannot be proven, can it be proven that such a number cannot exist? 

If such a triple existed with $a<b$, then $a,c,b$ would be an arithmetic progression of cubes.  But it is known that there cannot be three $n$-th powers in arithmetic progression if $n\geq 3$, see for example the paper of Darmon and Morel here.
There is probably an elementary proof of this when $n=3$, possibly in the paper of Dénes that they reference.
