Since Euclid prove that there is an infinite number of primes more than two millennia ago, a large number of mathematicians have been devising methods for finding large primes. Many have wasted obscene amounts of time in their search. Despite the development of several rules and theorems on factorization and primality testing over the years, they were still not enough. Thus, large and comprehensive books of factor tables were indispensable for research of prime numbers.
One of the first factor tables was published in Book of Calculations by Leonardo Fibonacci (c. 1175-c. 1250) in 1202. The book included a table of all the prime and composite numbers up to 100. In 1603, Pietro Cataldi (1548-1626) published a book entitled Treaties on Perfect Numbers which contained the factors of all natural numbers up to 750. In 1957, Fran van Schooten (1615-1660) listed all the prime numbers up to 9,929.
John Rahn (1622-1676) published the first extensive factor table in 1659. The table listed the prime factorizations of numbers up to 24,000. It was included as an appendix to an algebra book. Thomas Branker (1633-1676) extended Rahn’s table up to 100,000 in 1668. Johann Heinrich Lambert (1728-1777), who first proved that π is irrational, constructed a factor table for the first 102,000 positive integers.
Some were not so fortunate. After publishing his factor table, Lambert convinced Antonio Felkel (1740-1c. 800), a Viennese schoolteacher, to publish his own factor table. In 1776, Felkel published factor table for the numbers up to 408,000 at the expense of the Austrian Treasury. Unfortunately, due to a disappointing number subscribers, the Treasury recalled almost all the copies of the ill-fated books and converted the papers into cartridges to be used in a war against the Turks. That might have been one of the most ridiculous applications of mathematics to warfare.
In 1861, through the combined efforts of A. L. Crelle (1780-1855) and the lightning mental calculator Zacharias Dase (1824-1861), a factor table covering up to 9,000,000 was created and published in ten volumes.
The most significant achievement of this kind, however, goes to the factor table created by J. P. Kulik (1793-1863), a professor at the University of Prague. Kulik spent 22 years of his life to complete the factor table which listed the factors of the first 100 million positive numbers with the exceptions of the factors of 2, 3 and 5. He never published his work but he donated his manuscript, which filled several volumes, to the library at the University of Prague. Sadly, through someone’s carelessness, the second volume, which contained the factors of the numbers from 12,642,600 to 22,852,800, was lost.
The greatest factor table available was published by Derrick Norman Lehmer, an American mathematician (1867-1938) in 1910. The book is a cleverly designed single-volume table covering positive integers up to 10,000,000. Lehmer also pointed out several errors in Kulik’s work (which is unsurprising, considering that the entire work was only handwritten).
The rapid advancement of computing technology has significantly increased the efficiency of finding new primes. Nowadays, mathematicians have the luxury of enjoying the assistance of supercomputers and not to worry about doing the calculations manually.
As of the time of this writing, the largest known prime number is 274,207,281 − 1 (a number with 22,338,618 digits). It was found by the Great Internet Mersenne Primes Search (GIMPS) in January 2016.
Report of the Forty-Third Meeting of the British Association for the Advancement of Science: Held at Bradford in September 1873, Vol. 42, 1874
Scripta Mathematica, Vol. 4, 1936
Donald D. Spencer, Key Dates in Number Theory History, 1995
John Conway & Richard Guy, The Book of Numbers, 2012