Problem of the Week 1141

Capturing Five Integers

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For which positive real numbers x are there exactly 5 integers n such that the following statement is true?

x < n < x²

Source: A variation by me of a problem by Bill Sands in the Feb. issue of Crux Mathematicorum.

Extra credit: Given N, describe the set S(N) of real numbers x such that there are exactly N integers in the open interval determined by x and x².

© Copyright 2011 Stan Wagon. Reproduced with permission.