Journal of Integer Sequences, Vol. 19 (2016), Article 16.4.3 |

Hsing-Hua Senior High School

Miaoli 35144

Taiwan

**Abstract:**

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* ≡ *a*+*b* (mod 6);
moreover, we give a parametric function *r*_{i} →
(*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 ≤ *n* ≤
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*.

(Concerned with sequences A011541 A047696.)

Received April 29 2015; revised versions received November 17 2015; January 20 2016; March 28 2016.
Published in *Journal of Integer Sequences*, April 21 2016.

Return to