Editing Taxicab number (section)

==Cubefree taxicab numbers==
A more restrictive taxicab problem requires that the taxicab number be [[cubefree]], which means that it is not divisible by any cube other than 1<sup>3</sup>. When a cubefree taxicab number {{mvar|T}} is written as {{math|1=''T'' = ''x''<sup>3</sup> + ''y''<sup>3</sup>}}, the numbers {{mvar|x}} and {{mvar|y}} must be [[relatively prime]]. Among the taxicab numbers {{math|Ta(''n'')}} listed above, only {{math|Ta(1)}} and {{math|Ta(2)}} are cubefree taxicab numbers. The smallest cubefree taxicab number with three representations was discovered by [[Paul Vojta]] (unpublished) in 1981 while he was a graduate student:

<math display=block>\begin{align}
  15170835645 &= 517^3 + 2468^3 \\
  &= 709^3 + 2456^3 \\
  &= 1733^3 + 2152^3
\end{align}</math>

The smallest cubefree taxicab number with four representations was discovered by Stuart Gascoigne and independently by Duncan Moore in 2003:

<math display=block> \begin{align}
  1801049058342701083 &= 92227^3 + 1216500^3 \\
  &= 136635^3 + 1216102^3 \\
  &= 341995^3 + 1207602^3 \\
  &= 600259^3 + 1165884^3
\end{align}</math>
{{OEIS|id=A080642}}.