174 History of the Theory of Numbers. [Chap, vi 
m—r, while r digits precede the period. The condition that the length of 
the period be the maximum 0(6') is that 10 be a primitive root of h', whence 
5' = p", since 6'?^ 4 or 2^", p being an odd prime. 
P. Bachmann^°^ used a primitive root g of the prime p and set 
to the base g. We get the multiples Q, 2Q, . . . , (p — 1)Q by cyclic permuta- 
tion of the digits of Q. For p = 7, ^ = 10, Q = 142857. 
J. Kraus^"^ generaUzed the last result. When ri/n is converted into a 
periodic fraction to base g, prime to n, let ai, . . . , Ck be the quotients 
and ri, . . . , r^ the remainders. Then 
<7*-l 
rx = ax^^"^+ax+i/"^+- • -f«x-i (X = l,. • ■, k), 
n 
whence 
^x(aiS'*"^+ • • • +(ik) =n(«x9'*"^+ • ■ • 4-ax-i). 
In particular, let n be such that it has a primitive root g, and take ri = 1. 
Then 
ft 
and if rx is prime to n, the product ry,Q has the same digits as Q permuted 
cyclically and beginning with a^. 
H. Brocard^"^ gave a tentative method of factoring 10" — 1. 
J. Mayer^°^ gave conditions under which the period of z/P to base a, 
where z and a are relatively prime to P, shall be complete, i. e., corresponding 
digits of the two halves of the period have the sum a — 1. 
Heinrich Bork^°^ gave an exposition, without use of the theory of num- 
bers, of kno^n results on decimal fractions. There is here first published 
(pp. 36-41) a table, computed by Friedrich Kessler, showing for each prime 
p< 100000 the value of q={p — l)/e, where e is the length of the period 
for 1/p. The cases in which ^ = 1 or 2 were omitted for brevity. He 
stated that there are many errors in the table to 15000 by Reuschle.'*" 
Cunningham^^^ listed errata in Kessler's table. 
L. E. Dickson^"^ proved, without the use of the concept of periodic 
fractions, that every integer of D digits written to the base N, which is 
such that its products by D distinct integers have the same D digits in 
the same cyclic order, is of the form A{N^ — 1)/P, where A and P are 
relatively prime. A number of this form is an integer only when P is prime 
"iZeitschrift Math. Phys., 36, 1891, 381-3; Die Elemente der Zahlentheorie, 1892, 95-97. 
Alike discussion occurs in l'interm(5diaire des math., 5, 1898, 57-8; 10, 1903, 91-3. 
"'Zeitschrift Math. Phys., 37, 1892, 190-1. 
"'El Progreso Matematico, 1892, 25-27, 89-93, 114-9. Cf. rinterm^diaire des math., 2, 1895, 
323-4. 
'"Zeitschrift Math. Phys., 39, 1894, 376-382. 
losperiodische Dezimalbriiche, Progr. 67, Prinz Heinrichs-Gymn., Berlin, 1895, 41 pp. 
looQuart. Jour. Math., 27, 1895, 366-77. 
