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.

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On 15 Jan 2002, 23:55.