160 History of the Theory of Numbers. [Chap, vi 
of m terms. If \/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 10'" — 1 and another 
di\'ides 10" — 1. If \/a has a period of mnp terms, where vi, n, p are primes, 
but no factor has such a period, any factor of a divides 10"*- 1,. . ., or 
10"" — !. These theorems aid in factoring a. 
L. Euler^ gave numerical examples of the conversion of ordinary frac- 
tions into decimal fractions and the converse problem. 
Euler^'' noted that if 2p+l is a prime 40?2±1, ±3, ±9, ±13, it divides 
lO^-ljif 2p + l isaprime40;?±7, ±11, ±17, ±19, it divides 10^+1. 
Jean BernouUi^ gave a r^sum6 of the work by Wallis,^ Robertson,* 
Lambert^ and Euler,^ and gave a table showing the full period for 1/D for 
each odd prime D<200, and a like table when Z) is a product of two equal 
or distinct primes < 25. When the two primes are distinct, the table con- 
firms Wallis' 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 
l/D^, 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 ?n/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 Z) = 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 7^ = /)+ 162, 
the third set of 15 digits is found by adding unity to 7~p, etc.; since 7 
belongs to the exponent 12 modulo D, the period for 1/D contains 15-12 
digits. 
Jean Bernoulli^ 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 A-^30, except for k = ll, 17, 19, 23, 29, with doubt as to 
the primality of the largest factor when A' = 13, 15 or ^19. He stated 
(p. 325) erroneously^^ that (10^^ + l)/ll-23 has no factor <3000. Also, 
10''+1 = 7-1M3-211-9091-520S1. 
He gave part of the periods for the reciprocals of various primes ^601. 
L. Euler^^ wTote to Bernoulli concerning the latter's^ paper and stated 
criteria for the divisibility of 10^±1 by a prime 2p + l=4n±l. If both 
2 and 5 or neither occur among the divisors of n, n=F2, n=F6, then 10'' — 1 
is divisible by 2p-\-l. But if only one of 2 and 5 occurs, then 10^+1 is 
divisible by 2p+l [cf. Genocchi^^]. 
Henry Clarke^^ discussed the conversion of ordinary fractions into 
decimals without dealing with theoretical principles. 
'Algebra, I, Ch. 12, 1770; French trans!., 1774. 
'"Opusc. anal., 1, 1773, 242; Comm. Arith. Coll., 2, p. 10, p. 25. 
'Nouv. m6m. acad. roy. Berlin, ann^e 1771 (1773), 273-317. 
*Ibid., 318-337. 
"P. Seelhoff, Zeitschrift Math. Phys., 31, 1886, 63. Reprinted, Sphinx-Oedipc, 5, 1910, 77-8. 
"Nouv. m(Sm. acad. roy. Berlin, annde 1772 (1774), Histoire, pp. 35-36; Comm. Arith., 1, 584. 
^^he rationale of circulating numbers, London, 1777, 1794. 
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