A table of values of pi(x)

The following table contains values of pi(x), the number of prime numbers less than x :

p(x) = # { p x,    p is prime}.

For example, p(11) = 5 since there are 5 primes less than or equal to 11, which are 2, 3, 5, 7 and 11.

Each pi(x) value in this table has been checked, by a second different computation. In the bigger values, the check consisted in computing p(x+d) with d small compared to x, usually d = 107 or d = 108, with different parameters in the implementation, and by checking the difference with a classical sieve. The parity of p(x) was also checked on some values (see Henri Lifchitz page).

For an explanation of the third and fourth column, see the Li(x) discussion section.

x p(x) Li(x)-p(x) q(x) Comments
106 78,498 129 -0.209
107 664,579 336 -0.1809
108 5,761,455 753 -0.1339 Meissel 1871
109 50,847,534 1,700 -0.0994 Meissel 1886 (Corrected)
1010 455,052,511 3,103 -0.0594 Lehmer 1959 (Corrected)
1011 4,118,054,813 11,587 -0.0725
1012 37,607,912,018 38,262 -0.0781
1013 346,065,536,839 108,970 -0.0723 Bohmann 1972 (Corrected)
1014 3,204,941,750,802 314,889 -0.0678 Lagarias Miller Odlyzko 1985
1015 29,844,570,422,669 1,052,618 -0.0735 LMO 1985
1016 279,238,341,033,925 3,214,631 -0.0726 LMO 1985
2*1016 547,863,431,950,008 3,776,488 -0.0607 LMO 1985
4*1016 1,075,292,778,753,150 5,538,861 -0.0634 LMO 1985 (World record at that time, in 28 hours)
1017 2,623,557,157,654,233 7,956,588 -0.0581 Deléglise Rivat 1994
1.5*1017 3,893,882,469,583,623
1018 24,739,954,287,740,860 21,949,554 -0.05177 Deléglise Rivat 1994
2*1018 48,645,161,281,738,535 29,811,771 -0.0500 Deléglise 1996
3*1018 72,254,704,797,687,083 45,770,751 -0.0629 Deléglise 1996
4*1018 95,676,260,903,887,607 69,277,091 -0.0827 Deléglise 1996
5*1018 118,959,989,688,273,472 43,092,600 -0.0461 Deléglise 1996
6*1018 142,135,049,412,622,144 58,896,042 -0.0576 Deléglise 1996
7*1018 165,220,513,980,969,424 61,621,341 -0.0559 Deléglise 1996
8*1018 188,229,829,247,429,504 59,219,350 -0.0503 Deléglise 1996
9*1018 211,172,979,243,258,278 87,883,179 -0.0704 Deléglise 1996
1019 234,057,667,276,344,607 99,877,774 -0.0760 Deléglise 1996
2*1019 460,637,655,126,005,490 108,526,618 -0.0587 Deléglise 1996
3*1019 684,559,920,583,084,690 146,461,969 -0.0649 P. Demichel, X. Gourdon 2000 (1)
3.1*1019 706,849,401,776,940,275 146,921,468 -0.0641 P. Demichel, X. Gourdon 2000 (1)
4*1019 906,790,515,105,576,571 98,572,744 -0.0379 Deléglise 1996
5*1019 1,127,779,923,790,184,543 133,700,534 -0.0461 P. Demichel, X. Gourdon 2000 (1)
1020 2,220,819,602,560,918,840 222,744,643 -0.0546 Deléglise 1996 (World record at that time, in 11 days)
1.6*1020 3,516,585,752,930,430,595 278,427,046 -0.0542 P. Demichel, X. Gourdon 2001 Feb (1)
1.7*1020 3,731,397,829,842,461,168 206,233,344 -0.0389 P. Demichel, X. Gourdon 2001 Feb (1)
1.8*1020 3,945,938,652,811,699,917 237,941,373 -0.0437 P. Demichel, X. Gourdon 2001 Feb (1)
2*1020 4,374,267,703,076,959,271 472,270,046 -0.0823 X. Gourdon 2000 Nov
2.1*1020 4,588,082,544,160,859,769 336,320,666 -0.0572 P. Demichel, X. Gourdon 2001 Feb (1)
2.6*1020 5,654,086,442,526,321,042 485,136,141 -0.0743 P. Demichel, X. Gourdon 2001 Feb (1)
2.7*1020 5,866,733,862,193,360,875 283,701,595 -0.0427 P. Demichel, X. Gourdon 2001 Feb (1)
3*1020 6,503,696,293,016,202,398 479,876,029 -0.0686 P. Demichel, X. Gourdon 2000 Dec (1)
3.1*1020 6,715,709,842,660,010,419 359,287,610 -0.0505 P. Demichel, X. Gourdon 2001 Feb (1)
3.2*1020 6,927,578,466,326,617,308 475,518,185 -0.0658 P. Demichel, X. Gourdon 2001 Feb (1)
4*1020 8,617,821,096,373,621,600 272,942,796 -0.0338 P. Demichel, X. Gourdon 2000 Dec (1)
5*1020 10,720,710,117,789,005,897 387,506,247 -0.0430 P. Demichel, X. Gourdon 2000 Dec (1)
6*1020 12,814,731,195,053,369,962 ??? ??? P. Demichel, X. Gourdon 2001 Jan (1)
1021 21,127,269,486,018,731,928 597,394,253 -0.0471 X. Gourdon 2000 Nov
2*1021 41,644,391,885,053,857,293 1,454,564,714 -0.0816 pi(x) project, 2000 Dec
4*1021 82,103,246,362,658,124,007 1,200,472,717 -0.0479 pi(x) project, 2000 Dec
1022 201,467,286,689,315,906,290 1,932,355,207 -0.0491 pi(x) project, 2000 Dec
1.5*1022 299,751,248,358,699,805,270 2,848,114,312 -0.0592 P. Demichel, X. Gourdon, 2001 Feb (70 days on PIII 1 Ghz) (1)
2*1022 397,382,840,070,993,192,736 2,732,289,619 -0.0493 pi(x) project, 2001 Feb
4*1022 783,964,159,847,056,303,858 5,101,648,384 -0.0655 pi(x) project, 2001 Mar (Current world record)

(1) The computation was launched by Patrick Demichel with the program of Xavier Gourdon.


Back to

mathematical constants and computation


File translated from TEX by TTH, version 2.32.
On 15 Jan 2002, 23:55.