Sum of two cubes

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

in elementary algebra.

Binomial numbers generalize this factorization to higher odd powers.

Proof Starting with the expression, and multiplying by

distributing a and b over ,

and canceling the like terms,

Similarly for the difference of two cubes,

"SOAP" mnemonic The mnemonic "SOAP", short for "Same, Opposite, Always Positive", helps recall of the signs: {| cellspacing="4" |- style="text-align:center;line-height:0.9;font-size:90%;" | || originalsign || || Same || || Opposite || || AlwaysPositive |- | || 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;"|

! | ! | ! | !

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 or

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

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: or note a link between 1729 and 91: 19 is the total of the digits in 1729; 19 reversed is 91; 19 x 91 is 1729.

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

Sum of two cubes

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