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 A radical problem (Posted on 2016-11-15)
Let n be a 5-digit positive integer.
Given the infinitely nested radical expression below:

Y = sqrt(n+ sqrt(n+ sqrt(n+… sqrt(n+… ,-

What is the maximal value of n, such that Y is a positive integer?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 solution Comment 2 of 2 |
y^2 = n + y

y^2 - y - n = 0

y = (1 +/- sqrt(1 + 4*n)) / 2

sqrt(1 + 4*n) must be an odd integer for y to have a positive integer value.

Every y will have an n value: n = y^2 - y

The y^2 term will predominate. For n = 100000 (one larger than the largest 5-digit number, y would be around sqrt(100000) or  316.227766016838. So try y = 316, leading to n = 316^2 - 316 = 99540, a 5-digit number.  Just to check, y=317 yields 100172, a 6-digit number.

Of course we could do it the computer way:

Check all 5-digit n's:

For n = 10000 To 99999
disc = 1 + 4 * n
sr = Int(Sqr(disc) + 0.5)
If sr * sr = disc Then
If sr Mod 2 = 1 Then
Text1.Text = Text1.Text & n & Str((1 + sr) / 2) & crlf
End If
End If
DoEvents
Next

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 Posted by Charlie on 2016-11-15 10:39:36

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