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welshfella
 one year ago
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
 one year ago
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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welshfella
 one year ago
Best ResponseYou've already chosen the best response.2The result is also positive.

welshfella
 one year ago
Best ResponseYou've already chosen the best response.2I don't know if there is an analytical way of doing this . I did it by trial and error.

welshfella
 one year ago
Best ResponseYou've already chosen the best response.2I 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
 one year ago
Best ResponseYou've already chosen the best response.2so it was not all trial an error!

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3how do you know that the smallest number is positive ?

welshfella
 one year ago
Best ResponseYou've already chosen the best response.2I don't I assumed that.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3there can be some negative integers, which can be expressible as sum of two cubes in two different ways right

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3I am referring to your first reply in this thread :)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3\[u = a^3+b^3=c^3+d^3\]

welshfella
 one year ago
Best ResponseYou've already chosen the best response.2yes Well maybe i should change the question to lowest positive number.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Ahh you should change the question, because there is no "least" element in the set of integers.

welshfella
 one year ago
Best ResponseYou've already chosen the best response.2I also assumed (for no reason other than its the case with 1729) that one of the pairs of numbers differed by only 1)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.31729 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 ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3by trial and error i got 91 is that what you have too ?

welshfella
 one year ago
Best ResponseYou've already chosen the best response.2yes 3^3 + 4^3 = 91 6^3 + (5)^3 = 91

welshfella
 one year ago
Best ResponseYou've already chosen the best response.2I 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
 one year ago
Best ResponseYou've already chosen the best response.3Hey, 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
 one year ago
Best ResponseYou've already chosen the best response.3plugin \(a=1, b=0\) and we get \[\begin{align}(3*1^2+09)^3+(4*1^2−0+6=0)^3&= (6*1^2−0+0)^3\\&+(5*1^2+0+0)^3 \end{align}\] \[3^3+4^3 = 6^3+(5)^3\]
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