How many positive integers such that is less than 1000, make the following expression a natural number as well?
This is based off the Infinitely Nested Radical Problem
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Let n + n + n + ⋯ = k with k is a positive integer.
We have: n + k = k , which implies n = k 2 − k .
For posive integer k , we have 0 < k 2 − k < 1 0 0 0 when 2 ≤ k ≤ 3 2 .
So the final answer is 3 1 .