The $n$-th taxicab number, denoted $\text{Ta}(n)$, is the smallest integer that can be expressed as a sum of two positive integer cubes in $n$ different distinct ways.

$\text{Ta}(1) = 2 = 1^3 + 1^3$ is trivial, and the infamous $\text{Ta}(2) = 1729$ was known as early as the 17th century, much before the well-known Hardy-Ramanujan story.

$\text{Ta}(3)$ was found by [John Leech in 1957][1]. After no further discoveries for three decades, the quest for more taxicab numbers seems to have gained traction around the same time computer-assisted proofs became more widespread. [Rosenstiel, Dardis and Rosenstiel][2] found $\text{Ta}(4)$ in 1989; Dardis found $\text{Ta}(5)$ in 1994 and this was later confirmed by [Wilson in 1999][3]; and finally [Calude et al.][4] announced $\text{Ta}(6)$ in 2003 which was later verified by Hollerbach in 2008.

The best information we have regarding other taxicab numbers are the upper bounds for $\text{Ta}(7)$ through $\text{Ta}(12)$ provided by [Boyer in 2006-2008][5]. There seems to have been a relatively rapid succession in the discovery of taxicab numbers from early 1990s until mid-2000s. One would imagine, the quality of the computational tools we have access to nowadays would only have accelerated the search -- but the quest seems to be silent since Boyer's upper bounds. Why is this?


  [1]: https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/some-solutions-of-diophantine-equations/F8760DDDA9ADB1A2849F79362E494521
  [2]: http://www.rosenstiel.co.uk/cubes/index.htm
  [3]: https://web.archive.org/web/20050215201136/http://www.cs.uwaterloo.ca/journals/JIS/wilson10.html
  [4]: https://www.cs.auckland.ac.nz/~cristian/taxicab.pdf
  [5]: http://www.christianboyer.com/taxicab/