# On the Diophantine equation $a^3 + b^3 = c^3 + d^3$

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?

• $a=c=n^2, b=d=m^2$ is always a solution. So yes, they can all be perfect squares. Mar 31, 2023 at 18:54
• Mar 31, 2023 at 19:11
• @NickS, okay, but I think solutions with $a=c$ and $b=d$ can be regarded as "trivial". Mar 31, 2023 at 19:13
• All squares are unlikely as equal sums of 6th powers are believed to have at least 6 summands in total. Mar 31, 2023 at 19:19
• A complete parametric solution in positive integers of $a^3+b^3=c^3+d^3$ is known: see Choudhry 1998 Theorem 2. Apr 5, 2023 at 15:43

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.