Sum of two cubes

In mathematics, the sum of two cubes is a cubed number added to another cubed number.

Every sum of cubes may be factored according to the identity $${\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})}$$ in elementary algebra.¹

Binomial numbers generalize this factorization to higher odd powers.

Starting with the expression, ${\displaystyle a^{2}-ab+b^{2}}$ and multiplying by a + b¹ $${\displaystyle (a+b)(a^{2}-ab+b^{2})=a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2}).}$$ distributing a and b over ${\displaystyle a^{2}-ab+b^{2}}$,¹ $${\displaystyle a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}}$$ and canceling the like terms,¹ $${\displaystyle a^{3}+b^{3}.}$$

Similarly for the difference of cubes, $${\displaystyle {\begin{aligned}(a-b)(a^{2}+ab+b^{2})&=a(a^{2}+ab+b^{2})-b(a^{2}+ab+b^{2})\\&=a^{3}+a^{2}b+ab^{2}\;-a^{2}b-ab^{2}-b^{3}\\&=a^{3}-b^{3}.\end{aligned}}}$$

The mnemonic "SOAP", short for "Same, Opposite, Always Positive", helps recall of the signs:²³⁴

	original sign		Same		Opposite		Always Positive
a3	+	b3    =    (a	+	b)(a2	−	ab	+	b2)
a3	−	b3    =    (a	−	b)(a2	+	ab	+	b2)

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.⁵

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),⁶ 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) = 91,⁷ expressed as:

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

Cabtaxi(3), the smallest Cabtaxi number expressed in 3 different ways, is 4104,⁸ expressed as

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