floor function (AKA greatest integer function) and square roots

**hatsoff** · December 3rd 2009, 04:11 AM

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!

**qmech** · December 3rd 2009, 08:45 AM

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

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

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

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