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[–] 2 points3 points  (0 children)

This is a neat question. It's closely related to the irrationality measure of \pi, which is not a known quantity.

The relation is roughly as follows: a and b are about the same size, so you're really comparing the small number c to the small number 1/(ab), which is (up to a constant) about 1/b2. So if there were infinitely many pairs (a,b) such that d (as defined above) was less than 1, \pi would have irrationality measure at most 2 (and hence exactly 2).

Of course, you're asking whether there is a minimum. Since \pi is transcendental (and hence has irrationality measure \geq 2), for any t>2 there should only be finitely many (a,b) with (ab)t/2c < 1, so the minimum would certainly be attained. But unfortunately I don't think it's possible to move from this to the case t=2 (i.e., the case you're interested in) -- if there is a lower bound > 2 on the irrationality measure of \pi then the minimum would be attained (because there would only be finitely many pairs (a,b) with d<1) but a quick google check didn't indicate that such a lower bound has been proved.

[–] 1 point2 points  (1 child)

I took this list and calculated d for each approximation. The top three are:

a b c d
355 113 2.67e-7 0.0107012
22 7 1.26e-3 0.1947313
5419351 1725033 2.22e-14 0.2075797

Just a quick note- if you calculate d using the geometric mean of a and b rather than the product, 5419351/1725033 jumps into first place, but 355/113 gets bumped down only one spot to second place. It's a testament to just how damned efficient 355/113 is.

[–][S] 0 points1 point  (0 children)

Great empirical test. I guess there's no known analytical way to prove a minimum.

[–] 0 points1 point  (1 child)

Isn't this the same question that was on Reddit/Programming last week... http://www.reddit.com/r/programming/comments/10s2jg/all_rational_approximations_of_pi_are_useles/

[–][S] 0 points1 point  (0 children)

Slightly related.

[–]Combinatorics 0 points1 point  (1 child)

Edit: And of course, I'm wrong.

[–][S] 1 point2 points  (0 children)

Huh?
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