Skip to main content

Full text of "History Of The Theory Of Numbers - I"

See other formats

160                   HISTORY OF THE THEORY OF NUMBERS.              [CHAP, vi
of m terms. If I/a has a period of mn terms, where m and n are primes, while no factor has such a period, one factor of a divides 10m1 and another divides 10n1. If I/a has a period of mnp terms, where m, n, p are primes, but no factor has such a period, any factor of a divides 101,..., or 10np1. These theorems aid in factoring a.
L. Euler7 gave numerical examples of the conversion of ordinary fractions into decimal fractions and the converse problem.
Euler70 noted that if 2p+l is a prime 40n==l, 3, =*=9, =*= 13, it divides 10P-1; if 2p+l is a prime 40n7, 11, ==17, 19, it divides 10p-hl.
Jean Bernoulli8 gave a rsum6 of the work by Wallis,3 Robertson,5 Lambert6 and Euler,7 and gave a table showing the full period for 1/D for each odd prime D<200, and a like table when D is a product of two equal or distinct primes <25. When the two primes are distinct, the table confirms Wallis7 assertion that the length of the period for 1/D is the 1. c. m. of the lengths of the periods for the reciprocals of the factors. But for 1/D2, where D is a prime > 3, the length of the period equals D times that for 1/D. If the period for 1/D, where D is a prime, has D  1 digits, the period for m/D has the same digits permuted cyclically to begin with m. He gave (p. 310) a device communicated to him by Lambert: to find the period.for 1/D, where D = 181, we find the remainder 7 after obtaining the part p composed of the first 15 digits of the period; multiply l/D=p+7/D by 7; thus-the next 15 digits of the period are given by 7p; since 73 =D+162, the third set of 15 digits is found by adding unity to 72p, etc.; since 7 belongs to the exponent 12 modulo D, the period for 1/D contains 15-12 digits.
Jean Bernoulli9 made use of various theorems due to Euler which give the possible linear forms of the divisors of 10*=*= 1, and obtained factors of (10*-l)/9 when fc^SO, except for & = 11, 17, 19, 23, 29, with doubt as to the primality of the largest factor when fc=13, 15 or ^19. He stated (p. 325) erroneously10 that (10U-{-!)/! 1-23 has no factor <3000. Also,
1015+1 = 7-1M3-211-9091-52081.
He gave part of the periods for the reciprocals of various primes ^601.
L. Euler11 wrote to Bernoulli concerning the latter's9 paper and stated criteria for the divisibility of 10P=*=1 by a prime 2p-f-l=4n==l. If both 2 and 5 or neither occur among the divisors of ft, n=p2, nq=6, then 10P1 is divisible by 2p-\-l. But if only one of 2 and 5 occurs, then 10P+1 is divisible by 2p-f-l [cf. Genocchi39].
Henry Clarke12 discussed the conversion of ordinary fractions into decimals without dealing with theoretical principles.
'Algebra, I, Ch. 12, 1770; French transl., 1774.
7aOpusc. anal., 1, 1773, 242; Comm. Arith. Coll., 2, p. 10, p. 25.
Nouv. me*m. acad. roy. Berlin, anne 1771 (1773), 273-317.
'Ibid., 318-337.
10P. Seelhoff, Zeitschrift Math. Phys., 31, 1886, 63. Reprinted, Sphinx-Oedipe, 5, 1910, 77-8. uNouv. mem. acad. roy. Berlin, anne*e 1772 (1774), Histoire, pp. 35-36; Comm. Arith., 1, 584. "The rationale of circulating numbers, London, 1777, 1794.