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 10m—1 and another divides 10n—1. 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 10™—1,..., or 10np—1. 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 40n±7, ±11, =±=17, ±19, it divides 10p-hl. Jean Bernoulli8 gave a r£sum6 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 10P—1 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, ann£e 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.