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 $${\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})}$$ in elementary algebra. ^[1]

Binomial numbers generalize this factorization to higher odd powers.

Proof

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

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

"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
a3	+	b3    =    (a	+	b)(a2	−	ab	+	b2)
a3	−	b3    =    (a	−	b)(a2	+	ab	+	b2)

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

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

Ta(3), the smallest taxicab number expressed in 3 different ways, is 87,539,319, expressed as

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

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:

${\displaystyle 3^{3}+4^{3}}$ or ${\displaystyle 6^{3}-5^{3}}$

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

${\displaystyle 16^{3}+2^{3}}$, ${\displaystyle 15^{3}+9^{3}}$ or ${\displaystyle -12^{3}+18^{3}}$

See also

• Difference of two squares
• Binomial number
• Sophie Germain's identity
• Aurifeuillean factorization
• Fermat's last theorem

References

1. McKeague, Charles P. (1986). Elementary Algebra (3rd ed.). Academic Press. p. 388. ISBN   0-12-484795-1.
2. Kropko, Jonathan (2016). Mathematics for social scientists. Los Angeles, LA: Sage. p. 30. ISBN   9781506304212.
3. http://books.google.com/books?id=ppQ3DwAAQBAJ&pg=PA36
4. http://books.google.com/books?id=NKAFEAAAQBAJ&pg=PA62
5. 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.
6. "A001235 - OEIS". oeis.org. Retrieved 2023-01-04.
7. 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.
8. 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.

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.