## Monday, March 10, 2008

### Cubes, the Cabtaxi Number Fleet

A taxicab number is the smallest integer that is the sum of two positive cubes from n different sums. So there's one taxicab number for n = 1, 2, 3. . . A cabtaxi number is the smallest positive integer that is the sum of two positive or negative integers from n different sums. (See Wikipedia or Wolfram.) Only nine cabtaxi numbers are known, though more are supposed to exist even though they haven't been found yet. All but the first three are seven digits or more. It occurs to me that the numbers for real taxicabs should be just a few digits — more than one, but less than seven certainly. So I think it would be interesting to find all the two, three, or four digit candidates for cabtaxi numbers (the cabtaxi fleet) , and it shouldn't matter whether they're the smallest integer for n sums.

Therefore, a cabtaxi candidate is like a cabtaxi number except that it may not be the smallest integer; and, for this project I'll say that it should be from 2 to 4 digits. This set of cabtaxi candidates would include Cabtaxi(2) and Cabtaxi(3) — 91, 728 — as well as the taxicab number, 1729, which is Taxicab(2).

Here are a few of the candidates in the cabtaxi fleet, but probably not all:
91 = 63 – 53 = 33 + 43, Cabtaxi(2)
152 = 63 – 43 = 33 + 53
189 = 63 – 33 = 43 + 53
217 = 93 – 83 = 63 + 13
513 = 93 – 63 = 83 + 13
721 = 93 – 23 = 163 – 153
728 = 93 – 13 = 123 103 = 63 + 83, Cabtaxi(3)
1027 = 193 – 183 = 103 + 33
1729 = 13 + 123 = 93 + 103, Taxicab(2)
1736 = 183 – 163 = 123 + 23
3367 = 163 – 93 = 343 – 333
4104 = 183 – 123 = 153 + 93 = 163 + 23
5824 = 183 – 23 = 243 – 203
5859 = 193 – 103 = 273 – 243
7922 = 203 – 23 = 243 – 183
8216 = 383 – 363 = 203 + 63
8587 = 543 – 533 = 193 + 123

From this we should be able to develop a way to use cabtaxi candidates graphically, similar to the coprimes plots.

Two ways to look at Cabtaxi(2):