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 − a b + b 2 ) {\displaystyle 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 − a b + b 2 {\displaystyle a^{2}-ab+b^{2}} and multiplying by a + b ( a + b ) ( a 2 − a b + b 2 ) = a ( a 2 − a b + b 2 ) + b ( a 2 − a b + b 2 ) . {\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 a 2 − a b + b 2 {\displaystyle a^{2}-ab+b^{2}}, a 3 − a 2 b + a b 2 + a 2 b − a b 2 + b 3 {\displaystyle a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}} and canceling the like terms, a 3 + b 3 . {\displaystyle a^{3}+b^{3}.}

Similarly for the difference of cubes, ( a − b ) ( a 2 + a b + b 2 ) = a ( a 2 + a b + b 2 ) − b ( a 2 + a b + b 2 ) = a 3 + a 2 b + a b 2 − a 2 b − a b 2 − b 3 = a 3 − b 3 . {\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:

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 {\displaystyle 1^{3}+12^{3}} or 9 3 + 10 3 {\displaystyle 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 {\displaystyle 436^{3}+167^{3}}, 423 3 + 228 3 {\displaystyle 423^{3}+228^{3}} or 414 3 + 255 3 {\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, expressed as:

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

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

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