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176                         HlSTOBY OF THE THEOKY 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. Fujimaki114 noted that if 10m— 1 is exactly divisible by n, and the quotient is a^ . .am of m digits, the numbers obtained from the latter by cyclic permutations of the digits are all multiples of aa . . .am.
J. Cullen, D. Biddle, and A. Cunningham116 proved that the large factor of 14 digits of (1025+ 1)/(105+1) is a prime.
L. Kronecker116 treated periodic fractions to any base.
W. P. Workman117 corrected three errors in Shanks'61 table.
D. Biddle118 concluded erroneously that (1017— 1)/9 is a prime.
H. Hertzer119 extended Kessler's105 table from 100000 to 112400, noted Reuschle's40 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.124
P. Bachmann120 proved the chief results on periodic fractions and cyclic numbers to any base g.
A. Tagiuri121 proved theorems [F. Meyer,64 Perkins29] on purely periodic fractions to any base and on mixed fractions.
E. B. Escott122 noted a misprint in Bickmore's108 table and two omissions hi Lucas'91 table, but described inaccurately the latter table, as noted by A. Cunningham.123
A. Cunningham124 described various tables (cited above) which give the exponent to which 10 belongs, and listed many errata.
J. R. Akerlund126 gave the prime factors of 1 1 ... 1 (to n digits) for n^ 16, n=18.
K. P. Nordlund126 applied to periodic fractions the theorem that, if ni,. . ., nr are distinct odd primes, no one dividing a, then N = n1mi. . . nrmr divides afc— 1, where k=$(N)/2r~l. He gave the period of l/p for p a prime < 100 and of certain a/ p.
T. H. Miller,127 generalizing the fact that the successive pairs of digits hi the period for 1/7 are 14, 28, .. . , investigated numbers n to the base r for which
1   2n , 4n , Sn ,
u<Jour. of the Physics School in Tokio, 7, 1897, 16-21; Abh, Gesch. Math. Wiss., 28, 1910, 22.
118Math. Quest. Educat. Times, 72, 1900, 99-101.
116Vorlesungen uber Zahlentheorie, I, 1901, 428-437.
"'Messenger Math., 31, 1901-2, 115.
™Ibid., p. 34; corrected, ibid., 33, 1903-4, 126 (p. 95).
119Archiv Math. Phys., (3), 2, 1902, 249-252.
u°Niedere Zahlentheorie, I, 1902, 351-363.
m?eriodico di Mat., 18, 1903, 43-58.
mNouv. Ann. Math., (4), 3, 1903, 136; Messenger Math., 33, 1903-4, 49.
'"Messenger Math., 33, 1903-4, 95-96.
™Ibid., 145-155.
1MNyt Tidsskrift for Mat., Kjobenhavn, 16 A, 1905, 97-103.
"•Goteborgs Kungl. Vetenskaps-Handlingar, (4), VII-VIII, 1905.
mProc. Edinburgh Math. Soc., 26, 1907-8, 95-6.