164 History of the Theory of Numbers. [Chap, vi 
E. Catalan^" converted periodic decimals into ordinary fractions without 
using infinite progressions. When 1/13 is converted into a decimal, the 
period of remainders is 1, 10, 9, 12, 3, 4; repeat the period; starting in 
the series of 12 terms with any term (as 10), take the fourth term (4) after 
it, the fourth term (12) after that, etc.; then the sum 26 of the three is a 
multiple of 13. In general, if D is a prime and D — l=mn, the sum of n 
terms taken w by m in the period for N/D is a multiple of D [cf. Thibault'^]. 
If the sum of two terms of the period of remainders for N/D is D, the 
same is true of the terms following them. Hence the sum of corresponding 
terms of the two half periods is D. This happens if the number of terms 
of the period is <f){D). 
Thibault^^ denoted the numbers of digits in the periods for l/d and 
1/d' by m and m'. If d' is divisible by d, m' is divisible by m. If d and d' 
have no common prime factor other than 2 or 5, the number of digits in 
the period for \/dd' is the 1. c. m. of m, m'. Hence it suffices to know the 
length of the period for 1/p", where p is a prime. If 1/p has a period of m 
digits and if 1/p" is the last one of the series 1/p, 1/p^, . . . which has a 
period of m digits, then the period for 1/p" for a >n has mp"'^ digits. For 
p = 3, we have w = 2; hence 1/3'^ for r^2 has a period of y~^ digits. For 
any prime p for which 7^p^ 101, we have n = 1, so that 1/p" has a period 
of mp°-~^ digits. Note that \/p and 1/p^ have periods of the same length 
to base h if and only if h^~^ = 1 (mod p^). Proof is given of Catalan's^" first 
theorem, which holds only when 10"' ^1 (mod D), i. e., when m is not a 
multiple of the number of digits in the period. For example, the sum of 
the /cth and (6+A;)th remainders for 1/13 is not a multiple of 13. 
E. Prouhet^^ proved Thibault's" theorem on the period for l/p". He^^" 
noted that multiples of 142857 have the same digits permuted. 
P. Lafitte^^ proved Midy's^^ theorem that, if p is a prime not dividing 
m and if the period for m/p has an even number of digits, the sum of the 
two halves of the period is 9 ... 9. 
J. Sornin^^ investigated the number m of digits in the period for 1/Z), 
where D is prime to 10. The period is a; = (10"* - l)/D. First, let D = lOA: + 1 . 
Then x = \Qy — \, where 
10*"-^+ A; ,^ , , lO'-^-A:^ 
y = ^ = lOz+k, z = 
Finally, we reach v= \l — { — k)'^\/D, and x is an integer if and only if v 
is. Hence if we form the powers of the number k of tens in D, add 1 to 
the odd powers, but subtract 1 from the even powers of k, the first exponent 
giving a result divisible by D is the number m of digits in the period. 
»»Nouv. Ann. Math., 1, 1842, 464-5, 467-9. 
*nhid., 2, 1843, 80-89. 
"/bid., 5, 1846, 661. 
^IhU., 3, 1844, 376; 1851, 147-152. 
'Vbid., 397-9. Cf. Araer. Math. Monthly, 19, 1912, 130-2. 
w/Wd., 8, 1849, 50-57. 
