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Sum of two cubes

## Summary

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 ${\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})}$  in elementary algebra.[1]

Binomial numbers generalize this factorization to higher odd powers.

### "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.[2] 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.

${\displaystyle +}$  ${\displaystyle +}$  ${\displaystyle -}$  ${\displaystyle +}$  ${\displaystyle -}$  ${\displaystyle -}$  ${\displaystyle +}$ originalsign Same Opposite AlwaysPositive ${\displaystyle a^{3}}$ ${\displaystyle b^{3}\quad =\quad (a}$ ${\displaystyle b)(a^{2}}$ ${\displaystyle ab}$ ${\displaystyle b^{2})}$ ${\displaystyle a^{3}}$ ${\displaystyle b^{3}\quad =\quad (a}$ ${\displaystyle b)(a^{2}}$ ${\displaystyle ab}$ ${\displaystyle b^{2})}$

### Proof

Starting with the expression, ${\displaystyle a^{2}-ab+b^{2}}$  is multiplied by a and b[1] ${\displaystyle (a+b)(a^{2}-ab+b^{2})=a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2}).}$  By distributing a and b to ${\displaystyle a^{2}-ab+b^{2}}$ ,[1] ${\displaystyle a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}}$  and by canceling the alike 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}}}

## 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.[3]

## 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,[4] expressed as

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

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 ${\displaystyle 414^{3}+255^{3}}$

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,[5] expressed as:

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

The smallest Cabtaxi number expressed in 3 different ways is 4104,[6] expressed as

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