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

1. ## 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!

2. ## Examples

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
$