Sum of two cubes: Difference between revisions

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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.

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.^[1]

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,^[2] 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, expressed as:

3^{4}+4^{3}

or

6^{3}-5^{3}

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

16^{3}+2^{3}

,

15^{3}+9^{3}

or

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

References

^ 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.
^ 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.

@@ Line 22: / Line 22: @@
 [[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 [[Leonhard Euler|Euler]].<ref>{{Cite journal |last=Dickson |first=L. E. |date=1917 |title=Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers |url=https://www.jstor.org/stable/2007234 |journal=Annals of Mathematics |volume=18 |issue=4 |pages=161–187 |doi=10.2307/2007234 |issn=0003-486X}}</ref>
-== Taxicab numbers ==
+== Taxicab and Cabtaxi numbers ==
-[[Taxicab number]]s are numbers that can be expressed as a sum of two positive integer cubes in ''n'' distinct ways. The smallest taxicab number is 1729,<ref>{{Cite web |title=A001235 - OEIS |url=https://oeis.org/A001235 |access-date=2023-01-04 |website=oeis.org}}</ref> expressed as
+[[Taxicab number]]s 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,<ref>{{Cite web |title=A001235 - OEIS |url=https://oeis.org/A001235 |access-date=2023-01-04 |website=oeis.org}}</ref> expressed as
 :<math>1^3 +12^3</math> or <math>9^3 + 10^3</math>
@@ Line 29: / Line 29: @@
 :<math>436^3 + 167^3</math>, <math>423^3 + 228^3</math> or <math>414^3 + 255^3</math>
+[[Cabtaxi number]]s 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, expressed as:
-The smallest taxicab number expressed in 3 different ways by using both positive and negative integers is 4104,<ref name="tstc">{{Cite journal |last=Silverman |first=Joseph H. |date=1993 |title=Taxicabs and Sums of Two Cubes |url=https://www.jstor.org/stable/2324954 |journal=The American Mathematical Monthly |volume=100 |issue=4 |pages=331–340 |doi=10.2307/2324954 |issn=0002-9890}}</ref> expressed as
+:<math>3^4 + 4^3</math> or <math>6^3 - 5^3</math>
+The smallest Cabtaxi number expressed in 3 different ways is 4104,<ref name="tstc">{{Cite journal |last=Silverman |first=Joseph H. |date=1993 |title=Taxicabs and Sums of Two Cubes |url=https://www.jstor.org/stable/2324954 |journal=The American Mathematical Monthly |volume=100 |issue=4 |pages=331–340 |doi=10.2307/2324954 |issn=0002-9890}}</ref> expressed as
 :<math>16^3 + 2^3</math>, <math>15^3 + 9^3</math> or <math>(-12)^3+18^3</math>