Sum of two cubes

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. Proof Starting with the expression, a^2-ab+b^2 and multiplying by (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-ab+b^2, a^3 - a^2 b + ab^2 + a^2b - ab^2 + b^3 and canceling the like 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} "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 :1^3 +12^3 or 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, 423^3 + 228^3 or 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 or 6^3 - 5^3 Cabtaxi(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 == See also ==