Sum of two cubes

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

Visual proof of the formulas for the sum and difference of two cubes

Factorization

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.

Proof

Starting with the expression, ${\displaystyle a^{2}-ab+b^{2}}$ and multiplying by a + b^[1] $${\displaystyle (a+b)(a^{2}-ab+b^{2})=a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2}).}$$ 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, $${\displaystyle {\begin{aligned}(a-b)(a^{2}+ab+b^{2})&=a(a^{2}+ab+b^{2})-b(a^{2}+ab+b^{2})\\&=a^{3}+a^{2}b+ab^{2}\;-a^{2}b-ab^{2}-b^{3}\\&=a^{3}-b^{3}.\end{aligned}}}$$

"SOAP" mnemonic

The mnemonic "SOAP", short for "Same, Opposite, Always Positive", helps recall of the signs:^[2][3][4]

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

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.^[5]

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) = 1, 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 ${\displaystyle -12^{3}+18^{3}}$

See also

• Difference of two squares
• Binomial number
• Sophie Germain's identity
• Aurifeuillean factorization
• Fermat's last theorem