Sum of two cubes

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

Proof

Starting with the identity of 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

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

And by cancelling the alike terms, we get

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

The sum of two cubes does not result in a cube.

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 $(-12)^{3}+18^{3}$

History

This formula was first discovered by Srinivasa Ramanujan, in 1913, when G. H. Hardy received a letter from him. Hardy, instead of sending the letter back to Ramanujan, showed interest in the theorems that he never saw before and with a partner, J. E. Littlewood, he was able to prove some of these theorems.

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.