Chap. VI] PERIODIC DECIMAL FRACTIONS. 169 
Karl Broda^^ considered a periodic decimal fraction F having an even 
number r of digits in the period and a number m of p digits preceding the 
period. Let x be the first half of the period, y the second half. Then 
10"* 10"*+'" 10"'+^'" io"'+3^^- ■"10"' ' 10'"(102'"-1) 
_ 9(jp40'"+a;+p)+q 
9-10"'(10^+l) 
ifx+2/ = o(10'' — l)/9 = a...a(tor terms). The first paper treated the case 
p = m = 0, and gave the generalization to base a in place of 10: 
£ ,^ , _£ , a+(a-l)x -r , _ a'-l 
a^^a"-^a'''^-- ' " (a-l)(a'-+l) " ^+2/-^ ^_j- 
The case a = a — l shows that a purely periodic fraction to the base a equals 
(a:+l)/(a'"+l) if the sum of the half periods has all its digits (to base a) 
equal to a— 1. Returning to the base 10, and taking A'= 9(10'"H-1), 
Z = 9x+a, where each digit of x is ^a, we see that Z/N equals a decimal 
fraction in which x is the first half of the period of r digits, while the second 
half is such that the sum of corresponding digits in it and x is a. If R is the 
remainder after r digits of the period have been obtained, R-\-Z = a (10''+ 1). 
C. G. Reuschle^^ gave tables which serve to find numbers belonging to 
a given exponent < 100 with respect to a given prime modulus < 1000. 
P. Mansion''" gave a detailed proof that, if n is prime to 2, 3, 5, and if 
the period for 1/n has n — 1 digits, the sum of corresponding digits in the 
half periods is 9. 
T. Muir^^ proved that, if p is a prime, either of 
N' = 1 (mod f) , iV^P" = 1 (mod p''+") 
follows from the other. If Xi is the least positive integer x for which the 
first holds and if p' is the highest power of p dividing N""' — !, then Xip" is 
the least positive integer y for which N^ = l (mod p'+"). Hence the known 
theorem: If N=JIpi'^, where Pi,P2,-- are distinct primes, and if the period 
for \/pi has m^ digits, and if Pi' is the highest power of Pi dividing 10"^ — 1, 
the number of digits in the period for \/N is the 1. c. m. of the niip^^i'^*. He 
asked if 6 = 1 when p>3, as affirmed by Shanks.^^ 
Mansion's proof {ibid., 5, 1876, 33) by use of periodic decimals of the 
generalized Fermat theorem is quoted under that topic. 
D. M. Sensenig'^ noted that a prime p?^2, 5, divides iV if it divides the 
sum of the digits of N taken in sets of as many figures each as there are 
digits in the period for l/p. 
«»Archiv Math. Phys., 56, 1874, 85-98; 57, 1875, 297-301. 
"Tafeln complexer Primzahlen, Berlin, 1875. Errata by Cunningham, Mess. Math., 46, 
1916, 60-1. 
'"Nouv. Corresp. Math., 1, 1874-5, 8-12. 
"Messenger Math., 4, 1875, 1-5. 
"The Analyst, Des Moines, Iowa, 3, 1876, 25. 
i 
