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Thread: floor function (AKA greatest integer function) and square roots

  1. #1
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    floor function (AKA greatest integer function) and square roots

    Prove that of the two equations

    $\displaystyle \left\lfloor\sqrt{n}+\sqrt{n+1}\right\rfloor=\left \lfloor\sqrt{n}+\sqrt{n+2}\right\rfloor$,

    $\displaystyle \left\lfloor\sqrt[3]{n}+\sqrt[3]{n+1}\right\rfloor=\left\lfloor\sqrt[3]{n}+\sqrt[3]{n+2}\right\rfloor$

    the first holds for every positive integer $\displaystyle n$, but the second does not.

    I'm pretty much stuck on this one. Any help would be much appreciated. Thanks!
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  2. #2
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    Examples

    Can't help with the proof, but these might help in the 2nd part (from Excel):

    $\displaystyle
    \left\lfloor\sqrt[3]{15} + \sqrt[3]{16}\right\rfloor = 4
    $
    $\displaystyle
    \left\lfloor \sqrt[3]{15} + \sqrt[3]{17} \right\rfloor = 5
    $


    $\displaystyle
    \left\lfloor \sqrt[3]{42} + \sqrt[3]{43} \right\rfloor = 6
    $
    $\displaystyle
    \left\lfloor \sqrt[3]{42} + \sqrt[3]{44} \right\rfloor = 7
    $
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