<s>[INST] Integral solution to a³+3ab²=4c³ Please help if integral solutions to this equation exists or not
A³+3AB²=4C³
Such that A,B,C are disninct
I thought we may be possible to prove solutions exists if and only if A=B=C condition is satisfied.
 [/INST] Note that 
$$(A-B)^3=A^3-3A^2B+3AB^2-B^3 \\
(A+B)^3=A^3+3A^2B+3AB^2+B^3$$
This gives
$$A^3+3AB^2= \frac{1}{2} \left((A+B)^3+(A-B)^3 \right)$$
Therefore, if you have a solution
$$(A-B)^3+(A+B)^3=(2C)^3$$
Use FLT to deduce that all solutions satisfy $A=B$ or $A=-B$ or $A=0$.
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