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 So close, but no closer... (Posted on 2014-12-04)

Determine the value of the smallest positive integer N such that N1/N contains precisely one zero immediately following the decimal point.

Too easy?

 See The Solution Submitted by broll Rating: 3.0000 (2 votes)

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 computer assisted solution | Comment 1 of 5
A graphing calculator table shows

N     N^(1/N)
38  1.100457444033732
39  1.098491064028148

so part 1's answer is 39.

Part 2:

After narrowing it down somewhat via calculator, the following code, actually at the first iteration of i, finds N = 190660033.957375 produces 1.0000001.

n1 = 190660000#: v1 = n1 ^ (1 / n1)
n2 = 190670000#: v2 = n2 ^ (1 / n2)
For i = 1 To 50
frac = (1.0000001 - v1) / (v2 - v1)
n1 = n1 + (n2 - n1) * frac
v1 = n1 ^ (1 / n1)
Text1.Text = Text1.Text & n1 & Str(v1) & crlf
DoEvents
Next

To narrow it down further, I used UBASIC under DOSBox, with more precision than what is shown below:

Immediate mode of UBASIC says

N = 190660033 produces 1.0000001000000006368
N = 190660034 produces 1.0000001000000001398
N = 190660035 produces 1.0000000999999996428

Edited on December 4, 2014, 2:22 pm
 Posted by Charlie on 2014-12-04 14:21:37

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