If possible, find two positive integers such that:

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- December 24th 2008, 03:50 PMNonCommAlgAn inequality from Alex Lupas
If possible, find two positive integers such that:

- December 24th 2008, 11:33 PMbkarpuz
- December 25th 2008, 12:13 AMIsomorphism
Hey bkarpuz, I think he expects us to "find" m and n.

The inequality is pretty tight. I think m = 3391, n = 10630 works. But I did that the inelegant way (using MATLAB) (Crying)

I am wondering if a "Elementary and Middle School Math" technique exists for this (Thinking) - December 25th 2008, 12:15 AMNonCommAlg
- December 25th 2008, 05:59 AMbkarpuz
- December 26th 2008, 10:08 AMOpalg
According to my little pocket calculator (which may not be quite up to the job),

If then . Therefore .

So we are looking for a rational approximation to π that is less than π, but close enough that the difference is of the order of one-thousandth of the reciprocal of its denominator. Among the known rational approximations to π, the first one that seems to fit these conditions is . So take . Unless my little calculator is misleading me, that appears to work.

Those are bigger numbers than**Isomorphism**gets, but I don't have MATLAB to numbercrunch for me. - December 26th 2008, 12:35 PMNonCommAlg
let it's known that if then the sequence is dense in the unit interval. so there exists such that:

now let and then it's easy to see that this proves the existence of to find we need to find a solution for (Wondering)