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Math Help - 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

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

    \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 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
    Senior Member
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    Examples

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

    <br />
\left\lfloor\sqrt[3]{15} + \sqrt[3]{16}\right\rfloor  =  4 <br />
    <br />
\left\lfloor \sqrt[3]{15} + \sqrt[3]{17} \right\rfloor =  5 <br />


    <br />
\left\lfloor \sqrt[3]{42} + \sqrt[3]{43} \right\rfloor =  6 <br />
    <br />
\left\lfloor  \sqrt[3]{42} + \sqrt[3]{44} \right\rfloor =  7 <br />
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