Repeated Roots

Calculus Level 2

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

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

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

Joseph Newton by Joseph Newton
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