Chap. VI] PERIODIC DECIMAL FRACTIONS. 179 
Welsch^** discussed briefly the length of the period of a decimal fraction. 
B. Howarth^^^ noted that D^ is not a factor of (10^"-1)/(10'*-1) if D is 
a prime and n is not a multiple of the length of the period for 1/D. Again/^^ 
(^IQmnp^- _i)/9 is not divisible by (lO'"^-!) (10"^-1)/81. 
A. Cunningham^^^ factored 10^^ ± 1. Known factors of lO"^ 1 are given. 
Cunningham^^^ gave factors of 10"*^'* — !. 
A. Leman^^^ gave an elementary exposition and inserted proofs of Fer- 
mat's theorem and related facts, with the aim to afford a concrete introduc- 
tion to the more elementary facts of the theory of numbers. 
S. Weixer^^'' would compute the period P for 1/p by multiplication, 
beginning at the right. Let c be the final digit of P, whence pc = 10z — l. 
Then c is the first digit of the period P^ for z/p. The units digit Ci of 
cz = 10zi-\-Ci is the tens digit of P and the units digit of P^. In CiZ-\-Zi = 
1022+^2, C2 is the hundreds digit of P and the tens digit of P\ etc. 
A. Leman^^^ discussed the preceding paper. 
Problems^^^ on decimal fractions may be cited here. 
O. Hoppe^^^ proved that (10^^ — 1)/9 is a prime. 
M. Jenkins^^^ noted that if iV= a^6^. . . , where a,h,. . .are distinct primes 
9^2, 5, the period for 1/N is complementary (sum of corresponding digits of 
the half periods is 9) if and only if the lengths of the periods for 1/a, 1/b,. . . 
contain the same power of 2. 
Kraitchik^^^ of Ch. VII and Levanen^^ of Ch. XII gave tables of ex- 
ponents to which 10 belongs. Bickmore and Cullen^^^ of Ch. XIV factored 
10^^+1. 
Further Papers Involving No Theory of Numbers. 
J. L. Lagrange, Legons 61em. k I'^cole normale en 1795, Oeuvres 7, 200. 
James Adams, Annals Phil., Mag. Chem. (Thompson), (2), 2, 1821, 16-18. 
C. R. Telosius and S. Morck, Disquisitio. . . . Acad. Carolina, Lundae, 
1838 (in Meditationum Math. . . . PubHce Defendant C. J. D. Hill, 1831, 
Pt. II). 
J. A. Arndt, Archiv Math. Phys., 1, 1841, 101-4. 
J. Dienger, ibid., 11, 1848, 232; Jour, fur Math., 39, 1850, 67. 
Wm. Wiley, Math. Magazine, 1, 1882, 7-8. 
A. V. Filippov, Kagans Bote, 1910, 214-221 (pedagogic). 
i*^L'intermediaire des math., 21, 1914, 10. 
"sMath. Quest. Educat. Times, 28, 1915, 101-4. 
"»76id., 27, 1915, 33-4. 
^"Ibid., 29, 1916, 76, 88-9. 
i"Math. Quest, and Solutions, 3, 1917, 59. 
"'Vom Periodischen Dezimalbruch zur Zahlentheorie, Leipzig, 1916, 59 pp. 
""Zeitschrift Math. Naturw. Unterricht, 47, 1916, 228-230. 
i"/6id., 230-1. 
i"Zeitschrift Math. Naturw. Unterricht, 12, 1881, 431; 20, 188; 23, 584. 
i^Proc. London Math. Soc, Records of Meeting, Dec. 6, 1917, and Feb. 14, 1918, for a revised 
proof. 
i"Math. Quest. Educ. Times, 7, 1867, 31-2. Minor results, 32, 1880, 69; 34, 1881, 97-8: 37, 
1882, 44; 41, 1884, 113-4; 58, 1893, 108-9; 60, 1894, 128; 63, 1895, 34; 72, 1900, 75-6; 
74, 1901, 35; (2), 2, 1902, 65-6, 84-5; 4, 1903, 29, 65-7, 95; 7, 1905, 97, 106, 109-10; 8, 
1905, 57; 9, 1906, 73. Math. Quest, and Solutions, 3, 1917, 72 (table); 4, 1917, 22. 
