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发帖数: 4120 | 1 For a in Z+, f(x) = sin (a * pi /x )
then f'(x) = - cos ( a * pi /x ) * a * pi * (1/x/x)
For x in (1,a)
observe f(x) we conclude
(1) If the roots number of f(x) is odd, then a is a perfect sqare.
(2) If sin(sqrt(a) pi) < 0, f(x) has at least two roots.
(3) If sin(sqrt(a) pi) > 0, f(x) may have at least four roots or no roots.
you can search the least root <= a^{1/4}
If you have any suggested readings, let me know. no waste of time here. |
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