Let $a, b, c$ and $d$ be positive integers. What are the conditions that $a, b, c$ and $d$ should satisfy for the equality $$a^3 + b^3 = c^3 + d^3$$ to hold? In particular, can $a, b, c$ and $d$ be all perfect squares?

## 1 Answer

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You ask about the set of solutions for which $a,b,c,d$ are all perfect squares. This corresponds (with an abuse of notation) to solutions to the equation $$a^6 + b^6 = c^6 + d^6.$$ This defines a surface of general type. The Bombieri-Lang conjecture predicts that the solutions to the equation are not Zariski dense. Therefore the solutions to the original equation which are all perfect squares should be very rare indeed.

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