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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 Solution | Comment 1 of 2
Clearly the expression in the middle equals zero if n is a 4th power and will increase until n is the next power of 4.  So were looking for n to be 1 more than a power of 4: n=x^4+1 for whole number x.
This substitution gives us
(x4 + 1)1/4 - 1
A quick table shows the first value of x to be 30, which yields 9.26*10^-6

So the solution is n=30^4+1=810001

(It may be possible to avoid a table with some calculus.)

Edited on November 17, 2020, 10:48 am
  Posted by Jer on 2020-11-17 10:35:13

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