Sum 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.

Binomial numbers are the general of this factorization to higher odd powers.

The mnemonic "SOAP", standing for "Same, Opposite, Always Positive", is sometimes used to memorize the correct placement of the addition and subtraction symbols while factorizing cubes.^[1] When applying this method to the factorization, "Same" represents the first term with the same sign as the original expression, "Opposite" represents the second term with the opposite sign as the original expression, and "Always Positive" represents the third term and is always positive.

SOAP method
Input	Output	Same	Opposite and Always Positive
Input	Output	$a^{3}+b^{3}$	$(a+b)(a^{2}-ab+b^{2})$	$a^{3}+b^{3}$ → $(a+b)$	$a^{3}+b^{3}$ → $(a^{2}-ab+b^{2})$
$a^{3}-b^{3}$	$(a-b)(a^{2}+ab+b^{2})$	$a^{3}-b^{3}$ → $(a-b)$	$a^{3}-b^{3}$ → $(a^{2}+ab+b^{2})$

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.^[2]

Taxicab and Cabtaxi 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, after Ta(1), is 1729,^[3] 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}

Cabtaxi numbers are numbers that can be expressed as a sum of two positive or negative integers or 0 cubes in n ways. The smallest cabtaxi number, after Cabtaxi(1), is 91,^[4] expressed as:

3^{4}+4^{3}

or

6^{3}-5^{3}

The smallest Cabtaxi number expressed in 3 different ways is 4104,^[5] expressed as

16^{3}+2^{3}

,

15^{3}+9^{3}

or

-12^{3}+18^{3}

References

^ Kropko, Jonathan (2016). Mathematics for social scientists. Los Angeles, LA: Sage. p. 30. ISBN 9781506304212.
^ Dickson, L. E. (1917). "Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers". Annals of Mathematics. 18 (4): 161–187. doi:10.2307/2007234. ISSN 0003-486X.
^ "A001235 - OEIS". oeis.org. Retrieved 2023-01-04.
^ Schumer, Peter (2008). "Sum of Two Cubes in Two Different Ways". Math Horizons. pp. 8–9. Retrieved 2023-05-01.
^ Silverman, Joseph H. (1993). "Taxicabs and Sums of Two Cubes". The American Mathematical Monthly. 100 (4): 331–340. doi:10.2307/2324954. ISSN 0002-9890.