Sum of two cubes

Fermat's Last Theorem

Fermat's Last Theorem in the case of exponent 3 states that the sum of two non-zero integer cubes does not result in a non-zero integer cube. The first recorded proof of the exponent 3 case was given by Euler.

Taxicab and Cabtaxi numbers

A Taxicab number is the smallest positive number that can be expressed as a sum of two positive integer cubes in n distinct ways. The smallest taxicab number after Ta(1) = 2, is Ta(2) = 1729 (the Ramanujan number), expressed as

${\displaystyle 1^3+{12}^3}$ or ${\displaystyle 9^3+{10}^3}$

Ta(3), the smallest taxicab number expressed in 3 different ways, is 87,539,319, expressed as

${\displaystyle {436}^3+{167}^3}$, ${\displaystyle {423}^3+{228}^3}$ or ${\displaystyle {414}^3+{255}^3}$

A Cabtaxi number is the smallest positive number that can be expressed as a sum of two integer cubes in n ways, allowing the cubes to be negative or zero as well as positive. The smallest cabtaxi number after Cabtaxi(1) = 0, is Cabtaxi(2) = 91, expressed as:

${\displaystyle 3^3+4^3}$ or ${\displaystyle 6^3-5^3}$

note a link between 1729 and 91: 19 is the total of the digits in 1729; 19 reversed is 91; 19 x 91 is 1729.

Cabtaxi(3), the smallest Cabtaxi number expressed in 3 different ways, is 4104, expressed as

${\displaystyle {16}^3+2^3}$, ${\displaystyle {15}^3+9^3}$ or ${\displaystyle -{12}^3+{18}^3}$