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).

Pi digits

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

Pi digits mod 2

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).

The limit pi formulas

(4)

and

(5)

where is a Bernoulli
number (Plouffe 2022) can be used as a digit-extraction
algorithm for
(as well as ).
In particular, letting

(6)

the th digit to the right of the decimal point
of for is given by

(7)

where is the integer
part and
is the fractional part. Similar formulas can be
obtained using

(8)

and

(9)

where is an Euler
number, which gives a base-9 (or binary) digit-extraction
algorithm (Plouffe 2022). Related limits and formulas can also be obtained for
(Plouffe 2022).

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

(10)

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

(11)

(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, …
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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