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Microscopically small (Posted on 2020-11-17) Difficulty: 3 of 5
Find the smallest positive integer n for which 0 < n1/4 - [n1/4] < 10-5.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution re: Solution Comment 2 of 2 |
(In reply to Solution by Jer)

First, I wrote a program and found the same answer as Jer, 810001.

If you take the Taylor expansion of (n^4 + 1)^(1/4)
you get n^4^(1/4) + (1/4)*n^4^(-3/4)
or n + (1/4)*n^(-3)
Setting (1/4)*n^(-3) equal to 10^(-5)
results in n = the cube root of 25,000 = about 29.2401773821286

So the answer for n being 30 makes sense
30^4 + 1 = 810001

Edited on November 17, 2020, 3:53 pm
  Posted by Larry on 2020-11-17 15:52:42

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