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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?

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    $\begingroup$ $a=c=n^2, b=d=m^2$ is always a solution. So yes, they can all be perfect squares. $\endgroup$
    – Nick S
    Mar 31, 2023 at 18:54
  • $\begingroup$ See en.wikipedia.org/wiki/Taxicab_number $\endgroup$ Mar 31, 2023 at 19:11
  • $\begingroup$ @NickS, okay, but I think solutions with $a=c$ and $b=d$ can be regarded as "trivial". $\endgroup$
    – user501735
    Mar 31, 2023 at 19:13
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    $\begingroup$ All squares are unlikely as equal sums of 6th powers are believed to have at least 6 summands in total. $\endgroup$ Mar 31, 2023 at 19:19
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    $\begingroup$ A complete parametric solution in positive integers of $a^3+b^3=c^3+d^3$ is known: see Choudhry 1998 Theorem 2. $\endgroup$ Apr 5, 2023 at 15:43

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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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