mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 - ab + b^2)

elementary algebra Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics ...

. Binomial numbers generalize this

factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...

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
 (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

"SOAP" mnemonic

The

mnemonic A mnemonic device ( ), memory trick or memory device is any learning technique that aids information retention or retrieval in the human memory, often by associating the information with something that is easier to remember. It makes use of e ...

"SOAP", short for "Same, Opposite, Always Positive", helps recall of the signs: :

Fermat's last theorem

Fermat's last theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...

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 Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

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) = 1, is Ta(2) = 1729 (the Ramanujan number), expressed as :

1^3 +12^3

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

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

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

-12^3+18^3

Factorization

Proof

"SOAP" mnemonic

Fermat's last theorem

Taxicab and Cabtaxi numbers

See also

References

Further reading