CHAP, vi] PEKIODIC DECIMAL FRACTIONS. . 177 whence r2— 2n2 = 2. Besides the case r = 10, n = 7, he found r = 58, n=41, etc. A. Cunningham128 noted two errors in his paper113 and added 25212=l(mod 9972), 3901124=1 (mod 176) and cases modulo p2, where p = 103, 487, attributed to Th. Gosset. A. Cunningham129 gave tables of the periods of 1/N to the bases 2, 3, 5 for N^ 100. H. Hertzer130 noted three errors in Bickmore's108 table. A. G6rardin131 gave factors of 10n-l, n<100, and a table of the exponents to which 10 belongs modulo p, a prime < 10000, with a list of errors in the tables by Burckhardt and Desmarest. A. Filippov132 gave two methods of determining the generating factor for the periodic fraction for 1/b (cf. Lucas, The'orie des nombres, p. 178). G. C. Cicioni133 treated the subject. E. R. Bennett134 proved the standard theorems by means of group theory. W. H. Jackson136 noted that, if a is prime to 10 and if 6 is chosen so that 6<10, a6 = 10m— 1, the period for I/a may be written as &jl+10w+(10m)2+. . .+(10m)-1f -MO', where s is the exponent to which 10 belongs modulo a, and k is a positive integer. Thus for a = 39, 6 = 1, we have w=4, s = 6, and the period is 1+40+. . . + (40)5-M06, ^V = .02564i. G. Mignosi136 discussed the logic underlying the identification of an unending decimal with its generator p/q. A. Cunningham137 treated periodic decimals with multiples having the same digits permuted cyclically. F. Schuh138 considered the length qa of the period for l/p° for the base g, where p is a prime. He proved that qa is of the form g1pc, where 0^ eg a —2 when p = 2, a > 2, while O^c^a — 1 in all other cases. For a > 2, where q = q\ except when p — 2, g = 4m — 1, and then q = 2. Equality of periods for moduli pa and pb can occur for an odd prime p only when this period is qi, and for p = 2 only when it is 1 or 2. It is shown how to find the numbers g which give equal periods for pa and p, and the odd numbers g which give the period 2 for 2a. 128Math. Gazette, 4, 1907-8, 209-210. Sphinx-Oedipe, 8, 1913, 131. 129Math. Gazette, 4, 1907-8, 259-267; 6, 1911-12, 63-7, 108-116. 130Archiv Math. Phys., (3), 13, 1908, 107. »lSphinx-Oedipe, Nancy, 1908-9, 101-112. "'Spaczinskis Bote, 1908, pp. 252-263, 321-2 (Russian). u'La divisibilita del numeri e la teoria delle decimal! periodiche, Perugia, 1908, 150 pp. i"Amer. Math. Monthly, 16, 1909, 79-82. 185Annals of Math., (2), 11, 1909-10, 166-8. 138I1 Boll. Matematica Gior. Sc.-Didat., 9, 1910, 128-138. "'Math. Quest. Educat. Times, (2), 18, 1910, 25-26. liaXKannr A»/tVii'Af mi'alriivi/lA ^ Q 1 Q1 1 AHQ-AQQ r*.f S<*Tiuli 129-4 PV, VTT