# Repeated Roots

Calculus Level 2

There exists an expression with infinite nested roots that evaluates to an integer:

$3=\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}}}.$

How many integer values of $x$ between 1 and 1000 (inclusive) are there such that $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}}$ is also an integer?

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