Determine the smallest possible positive integer P which is not a perfect tenth power, but in the decimal expansion of its tenth root, the decimal point is followed by at least 14 consecutive zeroes.
We are looking for n, s.t. (n + ε)^10 = n^10 + 1
expand LHS
n^10 + 10(n^9)ε + O(ε^2) = n^10 + 1
10(n^9)ε = 1
n^9 = (10ε)^(-1)
n = (10ε)^(-1/9)
Suppose ε = 10^(-k) then 10ε = 10^(-(k-1))
n = 10^(-(k-1)*(-1/9))
Finally:
n = 10^((k-1)/9)
If k is 14, n is 10^(13/9) = 27.82559...
So try 27 and 28.
(27^10 + 1)^.1 = 27.0000000000000131137265... 13 zeros
(28^10 + 1)^.1 = 28.0000000000000094531754... 14 zeros
28^10 + 1 = 296196766695425
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Posted by Larry
on 2023-11-02 09:15:32 |
Analytic approach
