1729 is the smallest integer that can be written as the sum of 2 cubes of positive integers in 2 different ways. What is the smallest if negative integers are allowed?

Question

1729  is the smallest integer that can be written as the sum of 2 cubes of positive integers in 2 different ways. What is the smallest if negative integers are allowed?

welshfella · Answer

The result is also positive.

welshfella · Answer

I don't know if there is an analytical way of doing this . I did it by trial and error.

welshfella · Answer

I think its a fair assumption that 2 of the numbers will be close together and the other quite far apart as is the case with 1729

welshfella · Answer

so it was not all trial an error!

welshfella · Answer

maybe...

ganeshie8 · Answer

how do you know that the smallest number is positive ?

welshfella · Answer

I don't   I assumed that.

ganeshie8 · Answer

there can be some negative integers, which can be expressible as sum of two cubes in two different ways right

welshfella · Answer

yes

ganeshie8 · Answer

I am referring to your first reply in this thread :)

ganeshie8 · Answer

$$-|u| = a^3+b^3=c^3+d^3$$

welshfella · Answer

yes  Well maybe i should change the question to lowest positive number.

ganeshie8 · Answer

Ahh you should change the question, because there is no "least" element in the set of integers.

welshfella · Answer

I also assumed (for no reason other than its the case with 1729) that one of the pairs of numbers differed by only 1)

welshfella · Answer

yes

ganeshie8 · Answer

1729  is the smallest integer that can be written as the sum of 2 cubes of positive integers in 2 different ways. What is the smallest $\color{red}{\text{positive}}$ integer if negative integers are allowed?


does that look good ?

welshfella · Answer

yes

ganeshie8 · Answer

by trial and error i got 91
is that what you have too ?

welshfella · Answer

yes

3^3 + 4^3 = 91
6^3 + (-5)^3 = 91

welshfella · Answer

I guess there must be a lot of higher numbers which have the same property as 1729. I'll have to ask my grandson if he can find some. He is presently learning the programming language  Python in college . I expect that would find some.

ganeshie8 · Answer

Hey, I found this generating formula
$$\begin{align}(3a^2+5ab−5b^2)^3+(4a^2−4ab+6b^2)^3&= (6a^2−4ab+4b^2)^3\&+(-5a^2+5ab+3b^2)^3 \end{align}$$

ganeshie8 · Answer

plugin $a=1, b=0$ and we get
$$\begin{align}(3*1^2+0-9)^3+(4*1^2−0+6=0)^3&= (6*1^2−0+0)^3\&+(-5*1^2+0+0)^3 \end{align}$$

$$3^3+4^3 = 6^3+(-5)^3$$

ganeshie8 · Answer

http://math.stackexchange.com/questions/290635/algorithm-to-find-the-numbers-expressible-as-the-sum-of-two-positive-cubes-in-tw

welshfella · Answer

brilliant