176 History of the Theory of Numbers. [Chap.vi 
201 of Ch. VII]. He gave extensive tables, and references to papers on 
higher residues and to tables relating to period lengths. 
O. Fujimaki^^* noted that if 10'" — 1 is exactly divisible by n, and the 
quotient is Qi. . .a^ of jn digits, the numbers obtained from the latter by 
cycHc permutations of the digits are all multiples of Ci . . . a^. 
J. Cullen, D. Biddle, and A. Cunningham^ ^^ proved that the large factor 
of 14 digits of (10-^+l)/(10Hl) is a prime. 
L. Kronecker^^^ treated periodic fractions to any base. 
W. P. Workman^^^ corrected three errors in Shanks'^^ table. 
D. Biddle^^^ concluded erroneously that (10^^ — 1)/9 is a prime. 
H. Hertzer"' extended Kessler's^"'^ table from 100000 to 112400, noted 
Reuschle's'*'^ error on the conditions that 10 be a biquadratic residue of a 
prime p and gave the conditions that 10 be a residue of an 8th power 
modulo p. For errata in the table, see Cunningham. ^^ 
P. Bachmann^'° proved the chief results on periodic fractions and cyclic 
numbers to any base g. 
A. Tagiuri^^^ proved theorems [F. Meyer ,^ Perkins^^] on purely periodic 
fractions to any base and on mixed fractions. 
E. B. Escott^^^ noted a misprint in Bickmore's^^^ table and two omissions 
in Lucas'^^ table, but described inaccurately the latter table, as noted by 
A. Cunningham. ^^^ 
A. Cunningham^^^ described various tables (cited above) which give 
the exponent to which 10 belongs, and listed many errata. 
J. R. Akerlund^^° gave the prime factors of 11 ... 1 (to n digits) for n^ 16, 
n = 18. 
K. P. Nordlund^^^ applied to periodic fractions the theorem that, if 
Til, . . ., rir are distinct odd primes, no one dividing a, then N = ni"''. . . Ur""^ 
di\'ides a^ — l, where A: = 0(iV)/2'""\ He gave the period of \/p for p a 
prime < 100 and of certain a/p. 
T. H. Miller, ^-^ generalizing the fact that the successive pairs of digits 
in the period for 1/7 are 14, 28, ... , investigated numbers n to the base r 
for which 
1 _2n 4n 8n 
~ — 2"~1 4 I 6 ~r • • • ) 
n r r r" 
»"Jour. of the Physics School in Tokio, 7, 1897, 16-21; Abh. Gesch. Math. Wiss., 28, 1910, 22. 
i"Math. Quest. Educat. Times, 72, 1900, 99-101. 
"'Vorlesungen iiber Zahlentheorie, I, 1901, 428-437. 
"'Messenger Math., 31, 1901-2, 115. 
"«7&id., p. 34; corrected, ibid., 33, 1903^, 126 (p. 95). 
"•Archiv Math. Phys., (3), 2, 1902, 249-252. 
""Niedere Zahlentheorie, I, 1902, 351-363. 
"iPeriodico di Mat., 18, 1903, 43-58. 
»«Xouv. Ann. Math., (4), 3, 1903, 136; Messenger Math., 33, 1903-4, 49. 
'"Messenger Math., 33, 1903-4, 95-96. 
^^Ibid., 14.5-155. 
'"Nj-t Tidsskrift for Mat., Kjobenhavn, 16 A, 1905, 97-103. 
'"Goteborgs Kungl. Vetenskaps-Handlingar, (4), VII-VIII, 1905. 
"'Proc. Edinburgh Math. Soc, 26, 1907-8, 95-6. 
