Chap. VI] PERIODIC DECIMAL FRACTIONS. 177 
whence r^ — 2n^ = 2. Besides the case r = 10, n = 7, he found r = 58, n = 41 , 
etc. 
A. Cunningham^^* noted two errors in his paper"^ and added 
252^2 = ^j^o^ 9972>)^ 390112^ = 1 (mod 17«) 
and cases modulo p^, where p = 103, 487, attributed to Th. Gosset. 
A. Cunningham^^^ gave tables of the periods of \/N to the bases 2, 3, 5 
for N^ 100. 
H. Hertzer^^" noted three errors in Bickmore's^°* table. 
A. Gerardin^^^ gave factors of 10" — 1, n<100, and a table of the expo- 
nents to which 10 belongs modulo p, a prime < 10000, with a list of errors 
in the tables by Burckhardt and Desmarest. 
A. Filippov^^^ gave two methods of determining the generating factor 
for the periodic fraction for 1/6 (cf. Lucas, Th^orie des nombres, p. 178). 
G. C. Cicioni^^^ treated the subject. 
E. R. Bennett^^^ proved the standard theorems by means of group 
theory. 
W. H. Jackson^^^ noted that, if a is prime to 10 and if h is chosen so that 
h< 10, a& = 10m — 1, the period for \/a may be written as 
6]l + 10m+(10m)2+. . .+(10m)'-it -A;-10', 
where s is the exponent to which 10 belongs modulo a, and /c is a positive 
integer. Thus for a = 39, 6 = 1, we have m = 4, s = 6, and the period is 
1+40+. . . + (40)^-A:-10^ ^ = .025641. 
G. Mignosi^^^ discussed the logic underlying the identification of an 
unending decimal with its generator y/q. 
A. Cunningham^^^ treated periodic decimals with multiples having the 
same digits permuted cyclically. 
F. Schuh^^^ considered the length g^ of the period for 1/p" for the base g, 
where p is a prime. He proved that qa is of the form qip% where 0^ c^ a — 2 
when p = 2, a>2, while O^c^a — 1 in all other cases. For a>2, 
?a-l = giP""\ • • • , qa-c+1 = qiVj Qa-c = • • • = ?2 = ?, 
where q = qi txcept when p = 2, gr = 4m — 1, and then g = 2. Equality of 
periods for moduli p" and p'' can occur for an odd prime p only when this 
period is gi, 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 p" and p, and the odd numbers 
g which give the period 2 for 2". 
"8Math. Gazette, 4, 1907-8, 209-210. Sphinx-Oedipe, 8, 1913, 131. 
"9Math. Gazette, 4, 1907-8, 259-267; 6, 1911-12, 63-7, 108-116. 
"OArchiv Math. Phys., (3), 13, 1908, 107. 
"^Sphinx-Oedipe, Nancy, 1908-9, 101-112. 
"'Spaczinskis Bote, 1908, pp. 252-263, 321-2 (Russian). 
"'La divisibiht^ dei numeri e la teoria delle decimaU periodiche, Perugia, 1908, 150 pp. 
"«Amer. Math. Monthly, 16, 1909, 79-82. 
"'Annals of Math., (2), 11, 1909-10, 166-8. 
"'II Boll. Matematica Gior. Sc.-Didat., 9, 1910, 128-138. 
"^Math. Quest. Educat. Times, (2), 18, 1910, 25-26. 
"'Nieuw Archief Wiskunde, (2), 9, 1911, 408-439. Cf. Schuh,"'"*, Ch. VII. 
