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?

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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?

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

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so it was not all trial an error!
maybe...
how do you know that the smallest number is positive ?
I don't I assumed that.
there can be some negative integers, which can be expressible as sum of two cubes in two different ways right
yes
I am referring to your first reply in this thread :)
\[-|u| = a^3+b^3=c^3+d^3\]
yes Well maybe i should change the question to lowest positive number.
Ahh you should change the question, because there is no "least" element in the set of integers.
I also assumed (for no reason other than its the case with 1729) that one of the pairs of numbers differed by only 1)
yes
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 ?
yes
by trial and error i got 91 is that what you have too ?
yes 3^3 + 4^3 = 91 6^3 + (-5)^3 = 91
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.
Hey, I found this generating formula \[\begin{align}(3a^2+5ab−5b^2)^3+(4a^2−4ab+6b^2)^3&= (6a^2−4ab+4b^2)^3\\&+(-5a^2+5ab+3b^2)^3 \end{align}\]
plugin \(a=1, b=0\) and we get \[\begin{align}(3*1^2+0-9)^3+(4*1^2−0+6=0)^3&= (6*1^2−0+0)^3\\&+(-5*1^2+0+0)^3 \end{align}\] \[3^3+4^3 = 6^3+(-5)^3\]
http://math.stackexchange.com/questions/290635/algorithm-to-find-the-numbers-expressible-as-the-sum-of-two-positive-cubes-in-tw
brilliant

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