Sum of two cubes

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

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

Factorization

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

Binomial numbers generalize this factorization to higher odd powers.

Proof

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

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

"SOAP" mnemonic

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

	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.

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

$1^3+12^3$ or $9^3+10^3$

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

$436^3+167^3$, $423^3+228^3$ or $414^3+255^3$

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:

$3^3+4^3$ or $6^3-5^3$

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

$16^3+2^3$, $15^3+9^3$ or $-12^3+18^3$

See also

• Difference of two squares
• Binomial number
• Sophie Germain's identity
• Aurifeuillean factorization
• Fermat's Last Theorem