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

More Upper Bounds on Taxicab and Cabtaxi Numbers

Po-Chi Su
Hsing-Hua Senior High School
Miaoli 35144


For positive integers a, b and integers x, y such that S = a3 + b3 = x3 + y3, we prove that x+ya+b (mod 6); moreover, we give a parametric function ri → (x(ri),y(ri)) with (x(ri))3 + (y(ri))3 = a3 + b3 for chosen parameters ri, 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.

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(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.

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