Cubefree Taxicab Numbers
A more restrictive taxicab problem requires that the taxicab number be cubefree, which means that it is not divisible by any cube other than 13. When a cubefree taxicab number T is written as T = x3 + y3, the numbers x and y must be relatively prime for all pairs (x, y). Among the taxicab numbers Ta(n) listed above, only Ta(1) and Ta(2) are cubefree taxicab numbers. The smallest cubefree taxicab number with three representations was discovered by Paul Vojta (unpublished) in 1981 while he was a graduate student. It is
- 15170835645
- = 5173 + 24683
- = 7093 + 24563
- = 17333 + 21523.
The smallest cubefree taxicab number with 4 representations was discovered by Stuart Gascoigne and independently by Duncan Moore in 2003. It is
- 1801049058342701083
- = 922273 + 12165003
- = 1366353 + 12161023
- = 3419953 + 12076023
- = 6002593 + 11658843
(sequence A080642 in OEIS).
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