The Prime Glossary: tables of primes

tables of primes
(another Prime Pages' Glossary entries)

Glossary:

Prime Pages:

Top 5000:

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 l ist 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 le ast 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).

Notable tables of primes and least prime factors
(before electronic computers)
limit	who	when	type of table
800	Cataldi	1603	least prime factor
100,000	Brancker	1668	least prime factor
100,000	Kruger	1746	primes
102,000	Lambert	1770	least prime factor
408,000	Felkel	1776	least prime factor
400,031	Vega	1797	primes
1,020,000	Chernac	1811	least prime factor
3,036,000	Burkhardt	1816/17	primes
6,000,000	Crelle	1856	primes
9,000,000	Dase	1861	primes
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 numero us 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. (This is the longest such calculation that we know of, but surely a longer string of consecutive primes has been calculated by now!)

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)

References:

BS96
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)
Lehmer14
D. N. Lehmer, List of primes numbers from 1 to 10,006,721, Carnegie Institution Washington, D.C., 1914.
Ribenboim95
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.]