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Old 2017-02-27, 22:27   #10
Dr Sardonicus
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Feb 2017

2·2,687 Posts

Originally Posted by mickfrancis View Post
Having given some thought to this, I realise that what I really want is a way to find integer values for m such that
 <br />
(ceil(mr))^2- (mr)^2 < (mr)^s<br />
<br />

where s <= 1. I'm guessing this is a different kettle of fish...

This would appear to mean something like

| N/m - r| < c/m^(2 - s)

(c = positive constant), which is more easily achievable than having the exponent 2 in the denominator.

There is a notion of "best rational approximations," i.e. fractional approximations which are closer than any fraction with a smaller denominator. These consist of the "intermediate convergents" for the SCF. I know it's somewhere in Chrystal's Textbook of Algebra, but I'm sure something can be found on line.
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