Sum of two cubes

Visual proof of the formulas for the sum and difference 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

$a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$

in elementary algebra.

For the general of this factorization to higher odd powers see Binomial number#Factorization.

Proof

Starting with the expression, $a^{2}-ab+b^{2}$ is multiplied by a and b

$(a+b)(a^{2}-ab+b^{2})=a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2})$

By distributing a and b to $a^{2}-ab+b^{2}$, we get

$a^{3}-a^{2}b+ab^{2}+ba^{2}-ab^{2}+b^{3}$

And by cancelling the alike terms, we get

$a^{3}+b^{3}$

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 numbers

Taxicab numbers are numbers that can be expressed as a sum of two positive integer cubes in n distinct ways. The smallest taxicab number is 1729, expressed as

$1^{3}+12^{3}$ or $9^{3}+10^{3}$

The smallest taxicab number expressed in 3 different ways is 87,539,319, expressed as

$436^{3}+167^{3}$, $423^{3}+228^{3}$ or $414^{3}+255^{3}$

The smallest taxicab number expressed in 3 different ways by using both positive and negative integers is 4104, expressed as

$16^{3}+2^{3}$, $15^{3}+9^{3}$ or $(-12)^{3}+18^{3}$

Further reading

• Broughan, Kevin A. (January 2003). "Characterizing the Sum of Two Cubes" (PDF). Journal of Integer Sequences. 6 (4): 46. Bibcode:2003JIntS...6...46B.