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