Thaddeus Function

Algebra Level 4

Let
f(n)=11+2+12+3+13+41n+n+1.f(n) = \frac {1}{ \sqrt{1} + \sqrt{2}} + \frac {1}{\sqrt{2} + \sqrt{3}} + \frac {1}{\sqrt{3} + \sqrt{4}} \ldots \frac{1}{\sqrt{n}+\sqrt{n+1}}.

For how many positive integers nn, in the range 1n10001 \leq n \leq 1000 , is f(n)f(n) an integer?

This problem is posed by Thaddeus A.

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