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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?

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  1. welshfella
    • one year ago
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    The result is also positive.

  2. welshfella
    • one year ago
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    I don't know if there is an analytical way of doing this . I did it by trial and error.

  3. welshfella
    • one year ago
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    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

  4. welshfella
    • one year ago
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    so it was not all trial an error!

  5. welshfella
    • one year ago
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    maybe...

  6. ganeshie8
    • one year ago
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    how do you know that the smallest number is positive ?

  7. welshfella
    • one year ago
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    I don't I assumed that.

  8. ganeshie8
    • one year ago
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    there can be some negative integers, which can be expressible as sum of two cubes in two different ways right

  9. welshfella
    • one year ago
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    yes

  10. ganeshie8
    • one year ago
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    I am referring to your first reply in this thread :)

  11. ganeshie8
    • one year ago
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    \[-|u| = a^3+b^3=c^3+d^3\]

  12. welshfella
    • one year ago
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    yes Well maybe i should change the question to lowest positive number.

  13. ganeshie8
    • one year ago
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    Ahh you should change the question, because there is no "least" element in the set of integers.

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

  15. welshfella
    • one year ago
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    yes

  16. ganeshie8
    • one year ago
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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 \(\color{red}{\text{positive}}\) integer if negative integers are allowed? does that look good ?

  17. welshfella
    • one year ago
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    yes

  18. ganeshie8
    • one year ago
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    by trial and error i got 91 is that what you have too ?

  19. welshfella
    • one year ago
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    yes 3^3 + 4^3 = 91 6^3 + (-5)^3 = 91

  20. welshfella
    • one year ago
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    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.

  21. ganeshie8
    • one year ago
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    Hey, I found this generating formula \[\begin{align}(3a^2+5ab−5b^2)^3+(4a^2−4ab+6b^2)^3&= (6a^2−4ab+4b^2)^3\\&+(-5a^2+5ab+3b^2)^3 \end{align}\]

  22. ganeshie8
    • one year ago
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    plugin \(a=1, b=0\) and we get \[\begin{align}(3*1^2+0-9)^3+(4*1^2−0+6=0)^3&= (6*1^2−0+0)^3\\&+(-5*1^2+0+0)^3 \end{align}\] \[3^3+4^3 = 6^3+(-5)^3\]

  23. welshfella
    • one year ago
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    brilliant

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