tables of primes
Tables of primes have been kept for many centuries. The Ishango bone is an old bone (alternatively dated 6500 BC and 20,000 BC) which has three rows of notches. The middle row has groups of 11, 13, 17, and 19 notches. So this may be the oldest known list of primes.
We can be surer of the creators intent if we move closer to our time. The ancient Greeks definitely knew of primes (and Euclid proved that there were infinitely many of them), but the first practical "table of primes" we know of is a table of the least prime factors of the positive integers to 800. This table was created by Cataldi in 1603. Cataldi's table was soon followed by others. These tables helped the great mathematicians like Gauss and Legendre first guess the prime number theorem. They have been used to create many conjectures (and disprove many others).
|limit||who||when||type of table|
|800||Cataldi||1603||least prime factor|
|100,000||Brancker||1668||least prime factor|
|102,000||Lambert||1770||least prime factor|
|408,000||Felkel||1776||least prime factor|
|1,020,000||Chernac||1811||least prime factor|
|100,330,200||Kulik||1863 ?||least prime factor|
|10,007,000||D. N. Lehmer||1909||least prime factor|
|10,006,721||D. N. Lehmer||1914||primes|
Perhaps the most amazing of all tables ever created was Kulik's immense table. It was 8 volumes and 4212 pages. It had taken him nearly twenty years to complete! Sadly, the second of these volumes is now lost. (This loss is tempered by the numerous errors that the table contains.)
Unlike most of the previous tables, D. N. Lehmer's 1909 factor table was error free (other than the fact he consider one to be a prime). In 1914 he published a table of primes to the same limit. These tables were the first such tables to be widely available to mathematicians around the world. Most of the previous table existed only as a single copy stored in a mathematical archive.
Since the advent of computers, the need for large tables has been nearly erased. Using the sieve of Eratosthenes we can easily form a table of primes faster than they can be read from cards or a disk. So Baker and Gruenberger's table of primes to 104,395,289 on MicroCards (1959), and Bays & Hudson's table to 1,200,000,000,000 (1976) are no longer needed. When Brent needed the primes 4,400,000,000,000 in 1980, he just calculated them, then discarded them. (Far longer strings of consecutive primes has been calculated and discarded since then!)
Related pages (outside of this work)
- Lists of primes (that you can download)
- The Nth prime page (created by first forming a list of the primes below 1,000,000,000,000)
- E. Bach and J. Shallit, Algorithmic number theory, Foundations of Computing Vol, I: Efficient Algorithms, The MIT Press, Cambridge, MA, 1996. pp. xvi+512, MR 97e:11157 (Annotation available)
- D. N. Lehmer, List of primes numbers from 1 to 10,006,721, Carnegie Institution 1914. Washington, D.C.,
- P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]