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, providing n is greater than 1.
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, and I must limit n to 2 or more. 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
9728 = 243 – 163 = 203 + 123
This is still a very limited set, not enough for a large cab company. If I create a new category of products of cubes I can add a few more numbers to the fleet. These would be numbers that are the product of two positive cubes from n different products — I'll call them taxi products or taxi times:
64 = 13 × 43 = 23 × 23
216 = 13 × 63 = 23 × 33
512 = 13 × 83 = 23 × 43
729 = 13 × 93 = 33 × 33
1000 = 13 × 103 = 23 × 53
1728 = 33 × 43 = 23 × 63
2744 = 13 × 143 = 23 × 73
3375 = 13 × 153 = 33 × 53
4096 = 13 × 163 = 23 × 83 = 43 × 43
5832 = 13 × 183 = 23 × 93 = 33 × 63
8000 = 13 × 203 = 23 × 103 = 43 × 53
9261 = 13 × 213 = 33 × 73
Combining the cabtaxi candidates with the taxi products yields thirty numbers in the fleet of numbers with two to four digits. If the fleet needs to be larger, then I could always allow n to be one, which would increase the candidates to over 400.
"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." G. H. Hardy (1877 – 1947)
