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.^[1]

Binomial numbers generalize this factorization to higher odd powers.

Proof

Starting with the expression, $a^{2}-ab+b^{2}$ and multiplying by $a + b$ ^[1] $(a+b)(a^{2}-ab+b^{2})=a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2}).$ distributing a and b over $a^{2}-ab+b^{2}$ ,^[1] $a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}$ and canceling the like terms,^[1] $a^{3}+b^{3}.$

Similarly for the difference of cubes, ${\begin{aligned}(a-b)(a^{2}+ab+b^{2})&=a(a^{2}+ab+b^{2})-b(a^{2}+ab+b^{2})\\&=a^{3}+a^{2}b+ab^{2}\;-a^{2}b-ab^{2}-b^{3}\\&=a^{3}-b^{3}.\end{aligned}}$

"SOAP" mnemonic

The mnemonic "SOAP", short for "Same, Opposite, Always Positive", helps recall of the signs:^[2]^[3]^[4]

	original sign		Same		Opposite		Always Positive

$a 3$	$+$	$b 3 = (a$	$+$	$b)(a 2$	$-$	$ab$	$+$	$b 2)$
$a 3$	$-$	$b 3 = (a$	$-$	$b)(a 2$	$+$	$ab$	$+$	$b 2)$

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.^[5]

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),^[6] expressed as

1^{3}+12^{3}

or

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}

or

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,^[7] expressed as:

3^{3}+4^{3}

or

6^{3}-5^{3}

Cabtaxi(3), the smallest Cabtaxi number expressed in 3 different ways, is 4104,^[8] expressed as

16^{3}+2^{3}

,

15^{3}+9^{3}

or

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

References

^ ^a ^b ^c ^d McKeague, Charles P. (1986). Elementary Algebra (3rd ed.). Academic Press. p. 388. ISBN 0-12-484795-1.
^ Kropko, Jonathan (2016). Mathematics for social scientists. Los Angeles, LA: Sage. p. 30. ISBN 9781506304212.
^ http://books.google.com/books?id=ppQ3DwAAQBAJ&pg=PA36
^ http://books.google.com/books?id=NKAFEAAAQBAJ&pg=PA62
^ 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. JSTOR 2007234.
^ "A001235 - OEIS". oeis.org. Retrieved 2023-01-04.
^ Schumer, Peter (2008). "Sum of Two Cubes in Two Different Ways". Math Horizons. 16 (2): 8–9. doi:10.1080/10724117.2008.11974795. JSTOR 25678781.
^ 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. JSTOR 2324954.