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February 12

Proofs of rational bounds on π

In Proof that 22/7 exceeds pi a very pretty integral is given:

$$\underset{0}{\overset{1}{\int }}\frac{x^4{(1-x)}^4}{1+x^2}dx=\frac{22}{7}-\pi ,$$

which is evaluated via rather tedious polynomial long division, but simplifies nicely. $22/7$ is one of the continued fraction convergents to $\pi$, but the analogous integrals for the next convergents ($333/106$, $355/113$, etc.) are far more complicated—and they must be, since there must be a factor of 53 and 113 in each antiderivative, respectively. My question is simply whether this is a grand coincidence. The natural generalization (replacing the exponent 4 with $4n$, n>1) never seems to give a "good" approximation in Diophantine terms. Cheers, Ovinus (talk) 21:56, 12 February 2022 (UTC)

Are you asking whether it is a coincidence that the largest prime factor of the denominators of the next convergents is so much higher? A few steps later, in $104348/33215,$ we have $33215=5\times 7\times 13\times 73,$ so there is not some rule that the largest prime factor must keep increasing. In the continued-fraction expansion of $e,$ we even have a convergent $87/32.$  --Lambiam 23:49, 12 February 2022 (UTC)

Template:Ping Indeed there's no bound on the convergents' prime factors; I was just noting that for pi in particular, it would be impossible to achieve a value of, for example, $355/113-\pi$ with something like $\underset{0}{\overset{1}{\int }}P(x)/q(1+x^2)dx$, with $P$ being a polynomial with integer coefficients and $q,\mathrm{deg}P<112$. My question was more about intuition for why the long division magically simplifies to such a low common denominator; in the original integral, the antiderivative's coefficients' GCD is $1/1230$ or something like that.

But as a tangent, your example of $104348/33215$ makes me wonder if there are low-degree rational functions that could be used to obtain those approximations too. I'm sure there are, but I don't see a systematic way to find them. Ovinus (talk) 01:24, 13 February 2022 (UTC)

You should probably read the Stephen Lucas paper linked to in the article. (Amazingly, you don't have to go through a paywall to get to it.) I think the answer is yes, it is (more or less) a grand coincidence. The Lucas paper does mention that some "experimenting" was needed to produce integrals for 355/113 - pi, which I interpret as meaning that the method used would not be easily generalized. --RDBury (talk) 03:42, 13 February 2022 (UTC)