Integral solution to a³+3ab²=4c³

Question

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.

Answer 1

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 $A=0$.