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
    $\begingroup$ $a=c=n^2, b=d=m^2$ 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
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.