The following table contains values of pi(x), the number of prime numbers less than x :
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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.
mathematical constants and computation