Sum 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.

"SOAP" method
The mnemonic "SOAP", standing for "Same, Opposite, Always Positive", is sometimes used to memorize the correct placement of the addition and subtraction symbols while factorizing cubes. When applying this method to the factorization, "Same" represents the first term with the same sign as the original expression, "Opposite" represents the second term with the opposite sign as the original expression, and "Always Positive" represents the third term and is always positive.
 * {| cellspacing="4"

sign || || Same || || Opposite || || Always Positive !$$+$$ !$$+$$ !$$-$$ !$$+$$ !$$-$$ !$$-$$ !$$+$$ !$$+$$
 * - style="vertical-align:bottom;text-align:center;line-height:0.9;font-size:90%;"
 * || original
 * || style="border:1px solid;border-bottom:none;"|
 * || style="border:1px solid;border-bottom:none;"|
 * || style="border:1px solid;border-bottom:none;"|
 * || style="border:1px solid;border-bottom:none;"|
 * $$a^3$$
 * $$a^3$$
 * $$a^3$$
 * $$b^3\quad=\quad(a$$
 * $$b)(a^2$$
 * $$ab$$
 * $$b^2)$$
 * $$a^3$$
 * $$a^3$$
 * $$b^3\quad=\quad(a$$
 * $$b)(a^2$$
 * $$ab$$
 * $$b^2)$$
 * }

Proof
Starting with the expression, $$a^2-ab+b^2$$ is multiplied by a and b $$ (a+b)(a^2-ab+b^2) = a(a^2-ab+b^2) + b(a^2-ab+b^2). $$ By distributing a and b to $$a^2-ab+b^2$$, $$ a^3 - a^2 b + ab^2 + a^2b - ab^2 + b^3 $$ and by canceling the alike terms, a^3 + b^3

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

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
Taxicab numbers are numbers that can be expressed as a sum of two positive integer cubes in n distinct ways. The smallest taxicab number, after Ta(1), is 1729, expressed as
 * $$1^3 +12^3$$ or $$9^3 + 10^3$$

The smallest taxicab number expressed in 3 different ways is 87,539,319, expressed as
 * $$436^3 + 167^3$$, $$423^3 + 228^3$$ or $$414^3 + 255^3$$

Cabtaxi numbers are numbers that can be expressed as a sum of two positive or negative integers or 0 cubes in n ways. The smallest cabtaxi number, after Cabtaxi(1), is 91, expressed as:
 * $$3^3 + 4^3$$ or $$6^3 - 5^3$$

The smallest Cabtaxi number expressed in 3 different ways is 4104, expressed as
 * $$16^3 + 2^3$$, $$15^3 + 9^3$$ or $$-12^3+18^3$$