More upper bounds on taxicab and cabtaxi numbers

Su, Po-Chi

Journal of integer sequences, Tome 19 (2016) no. 4

Résumé

For positive integers $a, b$ and integers $x, y$ such that $S = a^{3} + b^{3} = x^{3} + y^{3}$, we prove that $x+y \equiv a+b$ (mod 6); moreover, we give a parametric function $r_{i} \to (x(r_{i}),y(r_{i}))$ with $(x(r_{i}))^{3} + (y(r_{i}))^{3} = a^{3} + b^{3}$ for chosen parameters $r_{i}$, and we conjecture that most such $S$ are multiples of 18 if $S$ is large enough. Accordingly, floating sieving is introduced and upper bounds on the Cabtaxi numbers $Ca(n)$ with $43 \le n \le 57$, and the Taxicab numbers $Ta(n)$ with $n = 23$,24 are given. Among them, $Ta(n)$ with $n = 23,24,$ and $Ca(n)$ with $n = 43,44,$ are included in the On-Line Encyclopedia of Integer Sequences.

Détail

Classification : 11D25
Keywords: taxicab number, cabtaxi number, magnification, splitting factor, sieving, floating sieving