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. 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 found $\text{Ta}(4)$ in 1989; Dardis found $\text{Ta}(5)$ in 1994 and this was later confirmed by Wilson in 1999; and finally Calude et al. 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. 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?

primitivesolutions to $x^3+y^3=m$, where primitive means $\gcd(x,y)=1$. Note that the taxicab numbers do not impose this gcd restriction. $\endgroup$J. London Math. Soc.28(1983), 1-7. OTOH, if we restrict to primitive solutions, then maybe we can get a uniform bound for the number of solutions in terms of the rank, even if $m$ is not cube-free. The key is to exploit the canonical height lower bound. $\endgroup$