170 History of the Theory of Numbers. [Chap, vi 
*A. J. M. Brogtrop^' treated periodic decimals. 
G. Bellavitis^* noted that the use of base 2 renders much more com- 
pact and convenient Gauss' ^^ table and hence constructed such a table. 
W. Shanks'^ found that the period for 1/p, where p = 487, is divisible 
by p, so that the period for 1/p^ has p — l digits. 
J. W. L. Glaisher^^ formed the period 05263. . . for 1/19 as follows: 
List 5; divide it by 2 and list the quotient 2; since the remainder is 1, 
divide 12 by 2 and list the quotient 6; divide it by 2 and list the quotient, 
etc. To get the period for 1/199, start with 50. To get the period, apart 
from the prefixed zero, for 1/49, start with 20 and divide always by 5; for 
1/499, start wath 200. 
Glaisher^^ noted that, if we regard as the same periods those in which 
the digits and their cyclic order are the same, even if commencing at differ- 
ent places, a number q prime to 10 will have/ periods each of a digits, where 
af=4){q). This was used to check Goodwyn's table. ^^ If g = 39, there are 
four periods each of six digits, li q — 1 belongs to the period for 1/q, the 
two halves of every period are complementary; if not, the periods form 
pairs and the periods in each pair are complementary. For each prime 
N< 1000, except 3 and 487, the period for l/N" has nA^*"^ digits if that 
for 1/iV has n digits. 
Glaisher'^^ collected various known results on periodic decimals and 
gave an account of the tables relating thereto. If q is prime to 10 and if 
the period for 1/q has (/)(g) digits, the products of the period by the 4>{q) 
integers <q and prime to q have the same digits in the same cyclic order; 
for example, if g = 49. He gave (pp. 204-6) for each g<1024 and prime 
to 10 the number a of digits in the period for 1/q, the number n of periods 
of irreducible fractions p/q, not regarding as distinct two periods having 
the same digits in the same cychc order, and, finally Euler's (f>(,q). The 
values of a and n were obtained by mere counting from the entries in Good- 
wyn's^^ "table of circles"; in every case, an = <j){q). For the prime p = 487, 
he gave the full periods for 1/p and 1/p", each of 486 digits, thus verifying 
Desmarest's^^ statement of the exceptional character of this p [cf. Shanks'^]. 
Glaisher^^ again stated the chief rules for the lengths of periods. 
The problem was proposed^" to find a number whose products by 2, . . . , 6 
have the same digits, but in a new order. 
Birger Hausted^^ solved this problem. Start with any number a of 
one digit, multiply it by any number p and let b be the digit in the units 
"Nieuw Archief voor VViskunde, Amsterdam, 3, 1877, 58-9. 
7«Atti Accad. Lincei, Mem. Sc. Fis. Mat., (3), 1, 1877, 778-800. Transunti, 206. See 62a 
of Ch. VII. 
"Proc. Roy. Soc. London, 25, 1877, 551-3. 
^^Messenger Math., 7, 1878, 190-1. Cf. Desmarest." 
"Report British Assoc, 1878, 471-3. 
"Proc. Cambridge Phil. Soc, 3, 1878, 185-206. 
"Solutions of the Cambridge Senate-House Problems and Riders for 1878, pp. 8-9. 
""Tidsskrift for Math., Kjobenhavn, 2, 1878, 28. 
"/bid., pp. 180-3. Jornal de Sc. Math, e Ast., 2, 1878, 154-6. 
