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. The end term of the identity, $b^{2}$, for the sum or difference of two cubes will always end in the addition of $b^{2}$.

Proof

Starting from the left-hand side, distribute $a^{2}-ab+b^{2}$ to $a+b$ to get

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

Using the distributive law, distribute a to $a^{2}-ab+b^{2}$ and b to $a^{2}-ab+b^{2}$ to get

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

By combining, both middle terms cancel:

$-a^{2}b+ba^{2}+ab^{2}-b^{2}a=0$

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

The identity does not actually equal a cube. In order to prove this, a and b must be a non-zero rational number. We will make $a=1$ and $b=2$. Plugging in a and b shows that

$1^{3}+2^{3}=(1+2)(1^{2}-1(2)+2^{2})$

Which if simplified shows that

$1+8=3(1-2+4)$

And simplifying the equation using the order of operations gets

$9=9$

9 is the resulting answer, although it is not a cube. However, it is a square, and so is $a=2$ and $b=2$, which equals to 16.

Taxicab numbers

Taxicab numbers are numbers that can be expressed as a sum of two cubes in 2, or more different ways. The smallest taxicab number is 1729, expressed as

$1^{3}+12^{3}$

and also

$9^{3}+10^{3}$.

The smallest number expressed as a sum of two cubes in 3 different ways require that both positive and negative integers are allowed. This number is 4104,expressed as

$16^{3}+2^{3}$

and

$15^{3}+9^{3}$

and also

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

If we don't allow negative integers, the smallest number expressed as a sum of two cubes in 3 different ways is 87,539,319, expressed as

$436^{3}+167^{3}$

and

$423^{3}+228^{3}$

and also

$414^{3}+255^{3}$.

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.