Pi Digits

 has decimal expansion given by

(1)

(OEIS A000796). The following table summarizes some record computations of the digits of .

	1999	Kanada, Ushio and Kuroda
	Dec. 2002	Kanada, Ushio and Kuroda (Peterson 2002, Kanada 2003)
	Aug. 2012	A. J. Yee (Yee)
	Aug. 2012	S. Kondo and A. J. Yee (Yee)
	Dec. 2013	A. J. Yee and S. Kondo (Yee)

The calculation of the digits of  has occupied mathematicians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his life calculating  to 35 places. Although he did not live to publish his result, it was inscribed on his gravestone. Wells (1986, p. 48) discusses a number of other calculations. The calculation of  also figures in the Season 2 Star Trek episode "Wolf in the Fold" (1967), in which Captain Kirk and Mr. Spock force an evil entity (composed of pure energy and which feeds on fear) out of the starship Enterprise's computer by commanding the computer to "compute to the last digit the value of pi," thus sending the computer into an infinite loop.

Al-Kāshi of Samarkand computed the sexagesimal digits of  as

(2)

(OEIS A091649) using -gons, a value accurate to 17 decimal places (Borwein and Bailey 2003, p. 107).

The binary representation of the decimal digits of  (top figure) and decimal representation (bottom figure) of  are illustrated above.

A plot of the first 1600 decimal digits of  (mod 2) is shown above (left figure), with the corresponding plot for 22/7 shown at right. Here, white indicates an even digit and black an odd digit (Pickover 2002, p. 285).

Spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140-141) and base-16 digit-extraction algorithms (the BBP formula) are known for . A remarkable recursive formula conjectured to give the th hexadecimal digit of  is given by , where  is the floor function,

(3)

 is the fractional part and  (Borwein and Bailey 2003, Ch. 4; Bailey et al. 2007, pp. 22-23).

Pi-primes, i.e., -constant primes occur at 2, 6, 38, 16208, 47577, 78073, 613373, ... (OEIS A060421) decimal digits.

The beast number 666 appears in  at decimals 2440, 3151, 4000, 4435, 5403, 6840, (OEIS A083625). The first occurrences of just  consecutive 6s are 7, 117, 2440, 21880, 48439, 252499, 8209165, 55616210, 45681781, ... (OEIS A096760), while  (or more) consecutive 6s first occur at 7, 117, 2440, 21880, 48439, 252499, 8209165, 45681781, 45681781, ... (OEIS A050285).

The digits 314159 appear at positions 176451, 1259351, 1761051, 6467324, 6518294, 9753731, 9973760, ... (correcting Pickover 1995).

The sequence 0123456789 occurs beginning at digits , , , , , and  (OEIS A101815; cf. Wells 1986, pp. 51-52).

The sequence 9876543210 occurs beginning at digits , , , , and  (OEIS A101816).

The sequence 27182818284 (the first few digits of e) occurs beginning at digit  (see also Pickover's sequence).

There are also interesting patterns for . 0123456789 occurs at , 9876543210 occurs at  and , and 999999999999 occurs at  of .

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of  (including the initial 3 and counting it as the first digit) are 33, 2, 7, 1, 3, 5, 8, 14, ... (OEIS A032445).

Scanning the decimal expansion of  until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 0, 68, 483, 6716, 33394, 569540, ... (OEIS A032510), which end at digits 33, 607, 8556, 99850, 1369565, ... (OEIS A080597).

A curiosity relating  to the beast number 666 involves adding the first three sextads of . First, note that

(4)

Now, skip ahead 15 decimal places and note that the sum is repeated as

(5)

(pers. comm., P. Olivera, Aug. 11, 2005; Olivera).

It is not known if  is normal (Wagon 1985, Bailey and Crandall 2001), although the first 30 million digits are very uniformly distributed (Bailey 1988).

The following distribution of decimal digits  is found for the first  digits of  (Kanada 2003). It shows no statistically significant departure from a uniform distribution.

	OEIS	2	3	4	5	6	7	8	9	10	11	12
0	A099291	8	93	968	9999	99959	999440	9999922	99993942	999967995	10000104750	99999485134
1	A099292	8	116	1026	10137	99758	999333	10002475	99997334	1000037790	9999937631	99999945664
2	A099293	12	103	1021	9908	100026	1000306	10001092	100002410	1000017271	10000026432	100000480057
3	A099294	11	102	974	10025	100229	999964	9998442	99986911	999976483	9999912396	99999787805
4	A099295	10	93	1012	9971	100230	1001093	10003863	100011958	999937688	10000032702	100000357857
5	A099296	8	97	1046	10026	100359	1000466	9993478	99998885	1000007928	9999963661	99999671008
6	A099297	9	94	1021	10029	99548	999337	9999417	100010387	999985731	9999824088	99999807503
7	A099298	8	95	970	10025	99800	1000207	9999610	99996061	1000041330	10000084530	99999818723
8	A099299	12	101	948	9978	99985	999814	10002180	100001839	999991772	10000157175	100000791469
9	A099300	14	106	1014	9902	100106	1000040	9999521	100000273	1000036012	9999956635	99999854780

The following table gives the first few positions at which a digit  occurs  times. The sequence 1, 135, 1698, 54525, 24466, 252499, 3346228, 46663520, 564665206, ... (OEIS A061073) given by the diagonal (plus any terms of the form 10 10's etc.) is known as the Earls sequence (Pickover 2002, p. 339). The sequence 999999 occurs at decimal 762 (which is sometimes called the Feynman point; Wells 1986, p. 51) and continues as 9999998, which is largest value of any seven digits in the first million decimals.

	OEIS	strings of 1, 2, ...s first occur at
0	A050279	32, 307, 601, 13390, 17534, 1699927, ...
1	A035117	1, 94, 153, 12700, 32788, 255945, ...
2	A050281	6, 135, 1735, 4902, 65260, 963024, ...
3	A050282	9, 24, 1698, 28467, 28467, 710100, ...
4	A050283	2, 59, 2707, 54525, 808650, 828499, ...
5	A050284	4, 130, 177, 24466, 24466, 244453, ...
6	A050285	7, 117, 2440, 21880, 48439, 252499, ...
7	A050286	13, 559, 1589, 1589, 162248, 399579, ...
8	A050287	11, 34, 4751, 4751, 213245, 222299, ...
9	A048940	5, 44, 762, 762, 762, 762, 1722776, ...

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Sloane, N. J. A. Sequences A000796/M2218, A032150, A032445, A035117, A036903, A036974, A037000, A037001, A037002, A037003, A037004, A037005, A037006, A037007, A037008, A048940, A050201, A050202, A050203, A050208, A050209, A050215, A050222, A050230, A050238, A050245, A050254, A050263, A050272, A050279, A050281, A050282, A050283, A050284, A050285, A050286, A050287, A060421, A061073, A080597, A083625, A091649, A096760, A099291, A099292, A099293, A099294, A099295, A099296, A099297, A099298, A099299, A099300, A101815, and A101816 in "The On-Line Encyclopedia of Integer Sequences."

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