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

Visual proof of the formulas for the sum and difference of two cubes

Factorization

edit

Every sum of cubes may be factored according to the identity   in elementary algebra.[1]

Binomial numbers generalize this factorization to higher odd powers.

"SOAP" method

edit

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.[2] 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.

original
sign
Same Opposite Always
Positive
                 
                 

Proof

edit

Starting with the expression,   and multiplying by a + b[1]   distributing a and b over  ,[1]   and canceling the like terms,[1]  .

Similarly for the difference of cubes,  

Fermat's last theorem

edit

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.[3]

Taxicab and Cabtaxi numbers

edit

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,[4] expressed as

  or  

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

 ,   or  

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,[5] expressed as:

  or  

Cabtaxi(3), the smallest Cabtaxi number expressed in 3 different ways, is 4104,[6] expressed as

 ,   or  

See also

edit

References

edit
  1. ^ a b c d 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. ^ 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.
  4. ^ "A001235 - OEIS". oeis.org. Retrieved 2023-01-04.
  5. ^ 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.
  6. ^ 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

edit