CHAPTER XIII. 
FACTOR TABLES. LISTS OF PRIMES. 
Eratosthenes (third century B.C.) gave a method, called the sieve or 
crib of Eratosthenes, of determining all the primes under a given limit I, 
which serves also to construct the prime factors of numbers <l. From 
the series of odd numbers 3, 5, 7, ... , strike out the square of 3 and every 
third number after 9, then the square of 5 and every fifth number after 25, 
etc. Proceed until the first remaining number, directly following that one 
whose multiples were last cancelled, has its square >l. The remaining 
numbers are primes. 
Nicomachus and Boethius^ began with 5 instead of with 5^, 7 instead of 
with 7^, etc., and so obtained the prime factors of the numbers <l. 
A table containing all the divisors of each odd number ^113 was printed 
at the end of an edition of Aratus, Oxford, 1672, and ascribed to Eratos- 
thenes by the editor, who incorrectly considered the table to be the sieve of 
Eratosthenes. Samuel Horsley^ believed that the table was copied by 
some monk in a barbarous age either from a Greek commentary on the 
Arithmetic of Nicomachus or else from a Latin translation of a Greek 
manuscript, published by Camerarius, in which occurs such a table to 109. 
Leonardo Pisano^ gave a table of the 21 primes from 11 to 97 and a 
table giving the factors of composite numbers from 12 to 100; to determine 
whether n is prime or not, one can restrict attention to divisors ^ ^/n. 
Ibn Albanna in his Talkhys^ (end of 13th century) noted that in using 
the crib of Eratosthenes we may restrict ourselves to numbers ^ -y/l. 
Cataldi^ gave a table of all the factors of all numbers up to 750, with a 
separate list of primes to 750, and a supplement extending the factor table 
from 751 to 800. 
Frans van Schooten® gave a table of primes to 9979. 
J. H. Rahn^ (Rhonius) gave a table of the least factors of numbers, not 
divisible by 2 or 5, up to 24000. 
T. Brancker^ constructed a table of the least divisors of numbers, not 
divisible by 2 or 5, up to 100 000. [Reprinted by Hinkley.^^] 
*Introd. in Arith. Nicomachi; Arith. Boethii, lib. 1, cap. 17 (full titles in the chapter on perfect 
numbers). Extracts of the parts on the crib, with numerous annotations, were given by 
Horsley.2 Cf. G. Bernhardy, Eratosthenica, Berlin, 1822, 173-4. 
2Phil. Trans. London, 62, 1772, 327-347. 
311 Liber Abbaci di L. Pisano (1202, revised 1228), Roma, 1852, ch. 5; Scritti, 1, 1857, 38. 
*Transl. by A. Marre, Atti Accad. Pont. Nuovi Lincei, 17, 1863-4, 307. 
"Trattato de' numeri perfetti, Bologna, 1603. Libri, Histoire des Sciences Math, en Italic, 
ed. 2, vol. 4, 1865, 91, stated erroneously that the table extended to 1000. 
«Exercitat. Math., libri 5, cap. 5, p. 394, Leiden, 1657. 
^Algebra, Zurich, 1659. WaUis,!"* p. 214, attributed this book to John Pell. 
*An Introduction to Algebra, translated out of the High-Dutch [of Rahn's' Algebra] into 
EngHsh by Thomas Brancker, augmented by D. P. [=Dr. Pell], London, 1668. It is 
cited in Phil. Trans. London, 3, 1668, 688. The Algebra and the translation were de- 
scribed by G. Wertheim, BibUotheca Math., (3), 3, 1902, 113-126. 
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