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?
by
Joseph Newton