CHAP, vi] PEEIODIC DECIMAL FRACTIONS. 179
Welsch144 discussed briefly the length of the period of a decimal fraction.
B. Howarth145 noted that D2 is not a factor of (10^-1)/(10W-1) if D is a prime and n is not a multiple of the length of the period for 1/D. Again,146 (l(Tnp2 -l)/9 is not divisible by (1(TP-1) (10np~l)/81.
A. Cunningham147 factored 10"=*= 1. Known factors of 10n=*= 1 are given.
Cunningham148 gave factors of 10mpn — 1.
A. Leman149 gave an elementary exposition and inserted proofs of Fer-mat's theorem and related facts, with the aim to afford a concrete introduction to the more elementary facts of the theory of numbers.
S. Weixer150 would compute the period P for l/p by multiplication, beginning at the right. Let c be the final digit of P, whence pc = 102 — 1. Then c is the first digit of the period P1 for z/p. The units digit GI of cz = 10zi-f-Ci is the tens digit of P and the units digit of P1. In clz-\-zl~ 10z2+c2, C2 ig ^ne' hundreds digit of P and the tens digit of P1, etc.
A. Leman151 discussed the preceding paper.
Problems152 on decimal fractions may be cited here.
0. Hoppe153 proved that (1019~l)/9 is a prime.
M. Jenkins154 noted that if N= apb9..., where &,&,.. .are distinct primes 5^2, 5, the period for l/N is complementary (sum of corresponding digits of the half periods is 9) if and only if the lengths of the periods for I/a, 1/b,... contain the same power of 2.
Kraitchik125 of Ch. VII and Levanen37 of Ch. XII gave tables of exponents to which 10 belongs. Bickmore and Cullen115 of Ch. XIV factored 1025+1.
FURTHER PAPERS INVOLVING No THEORY OF NUMBERS.
J. L. Lagrange, Lemons e*16m. a l'£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. . . . Publice 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. Pilippov, Kagans Bote, 1910, 214-221 (pedagogic).
144L'interm6diaire des math., 21, 1914, 10.
145Math. Quest. Educat. Times, 28, 1915, 101-4.
146/6id., 27, 1915, 33-4.
™Ibid., 29, 1916, 76, 88-9.
149Math. Quest, and Solutions, 3, 1917, 59.
149Vom Periodischen Dezimalbruch zur Zahlentheorie, Leipzig, 1916, 59 pp.
160Zeitschrift Math. Naturw. Unterricht, 47, 1916, 228-230.
^Ibid., 230-1.
18JZeitschrift Math. Naturw. Unterricht, 12, 1881, 431; 20, 188; 23, 584.
lB3Proc. London Math. Soc., Records of Meeting, Dec. 6,1917, and Feb. 14, 1918, for a revised
proof. »<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.