Approximation of pi

Approximation of π by rationals of the form a/b² .

Oleg Karpenkov conjectured that any real number ξ can be approximated by a rational of the form a/b² such that the inequality

| ξ - a/b²| < c/b³,
has infinitely many solutions in integers a and b [1] (see also Hardy and Littlewood [3], Heilbronn [4] and Zaharescu [5]).
The table below shows the values of b corresponding to approximations of π=3.14159 ... that are less in absolute value than 1/b³. The value of a can be computed as the nearest integer to πb². The table was computed by means of an algorithm [2] that performs b^1/2 polynomial time operations in average to compute the items in it.
A PARI script to calculate the convergents of the form a/b² for any real can be found here.

#	b	#	b	#	b
0	1
1	2	36	207207	71	765799023035418
2	3	37	265280	72	924764108790473
3	6	38	537766	73	1063349625535625
4	7	39	925036	74	1180700047859443
5	8	40	1693817	75	1392877612852012
6	13	41	1803181	76	2265120792743544
7	28	42	2053555	77	5655090457146592
8	32	43	11518526	78	10791046500563708
9	41	44	66797547	79	29523781971207577
10	50	45	90328940	80	29610072444366717
11	81	46	99409371	81	71502903187242438
12	85	47	105345415	82	199860004524767978
13	113	48	201049505	83	227906317501081783
14	141	49	343594795	84	399720009049535956
15	198	50	654075889	85	438348720358029255
16	267	51	934979323	86	526976478335568535
17	659	52	1061368874	87	530420552026510882
18	1014	53	10007486517	88	599580013574303934
19	1445	54	10143034860	89	775069693146270992
20	1650	55	17098866126	90	809695991456163835
21	1738	56	30663791044	91	2040782552597732857
22	2028	57	242095640151	92	2256217400888336760
23	2163	58	817685025481	93	2603652578578390773
24	2658	59	1100586602840	94	3397660065732068041
25	3091	60	1223837592174	95	14321981424900356103
26	3144	61	10033284088464
27	10224	62	29365202693872
28	13271	63	35649558528135
29	13451	64	45779483485139
30	18722	65	55318927739742
31	23871	66	73713181052162
32	26902	67	101266519548801
33	40353	68	103747516485092
34	100525	69	254245202820313
35	180850	70	326784622391234

[1] Oleg Karpenkov "Approximating reals by rationals of the form a/b²"

[2] I. Jimenez Calvo, "An algorithm to approximate reals by rationals of the form a/b²".

[3] G. H. Hardy and J. E. Littlewood. "Some problems of diophantine approximation I: The fractional part of n^k&theta". Acta Math. 37, 155-191 (1914).

[4] H. Heilbronn. "On the distribution of the sequence n²&theta (mod 1)". Quart. J. Math. Oxford Ser. 19, 249-256 (1948).

[5] A. Zaharescu. "Small values of n²&alpha". Invent. Math. 121, 379-388, (1995).

Last updated: April, 13, 2007

Homepage of Ismael Jiménez Calvo