178 History of the Theory of Numbers. [Chap, vi 
T. Ghezzi^^^ considered a proper irreducible fraction m/p with p prime 
to the base b of numeration. Let h belong to the exponent n modulo p. In 
7nb = pqi-\-ri, rih = pq2+r2,. . ., 0<ri<p, 0<r2<p,. . ., 
fi,. . ., r„ are distinct and r„ = w. Multiply the respective equations by 
6""^, 6""^,. . . and add; we see that 
p hr-i 
A similar proof shows that m/p equals a fraction ^xith. the denominator 
b'{b'* — l) when 6 = aia2a3, p^piCL^a^a^^ the a's being primes and pi rela- 
tively prime to b, while 6' is the least power of b having the di\isor ai^a^a^, 
and n is the exponent to which b belongs modulo pi. 
F. Stasi^*° gave a long proof showing that the length of the period for 
b/a does not exceed that for 1/a. If the period A for 1/p has m digits and 
n = p5 is prime to 10, the length of the period for \/n is m if A is divisible 
by q; is mi if A is prune to q and if the least A(10'"^*"^^+ . . . +1) divisible 
by q has m = i; and is mj if A=A'a, q = aq', with A', q' relatively prime, 
while the least A' (10"'^*'-^^+ ... +1) divisible by q' has k=j. For a prime 
p5^2, 5, let 
1 A 
h 
i 
p'' 10"* -1' 
and let A^ be the first of the periods of successive powers of 1/p not divisible 
by p; then the period for l/p''+^' has wp^' digits. If p, is a prime 9^2, 5, 
and Ti is the length of the period for l/p„ and if l/pj^< is the highest power 
of 1/pi with a period of Ti digits, the length of the period for l/p,"* is 
T-' = r{pi''i~^i and that for l/II Pi"< is a multiple of the 1. c. m. of the r/. 
If n is prime to 10 and if ri, . . . , r;„ = 1 are the successive remainders on 
reducing \/n to a decimal, then r^=r2i (mod n). Hence if 1/n has a period 
of 2i digits, r^ = \ (mod n) and conversely. But if it has a period of 
2i+l digits, r-+i = 10 and conversely. 
*K. W. Lichtenecker^^^ gave the length of the period for 1/p, when p is 
a prime ^307, and the factors of 10^ — 1, r?^ 10. 
L. Pasternak^^^ noted that, after multiplying the terms of a fraction by 
9, 3 or 7, we may assume the denominator iV' = 10m — 1. To convert Rq/N 
into a decimal, we have 10Rk-i = Nyk+Rk (^ = 1, 2, . . .). Set 7?^. = lOz^+e^t, 
ek^ 9. Since Vk'^ 9, e^ = Vk and Rk-i = mCk+Zk. Hence the successive digits 
of the period are the unit digits of the successive remainders. 
E. Maillet^^^ defined a unique development Oo+ai/n+ 02/11^+ ... of an 
arbitrary number, where the Oi are integers satisfjdng certain conditions. 
He studied the conditions that the development be limited or periodic. 
"»I1 Boll. Matematica Gior. Sc.-Didat., 9, 1910, 263-9. 
""/bid., 11, 1912, 226-246. 
i"Zeit3chr. fur das Realschulwesen, 37, 1912, 338-349. 
i^L'enseignement math., 14, 1912, 285-9. 
»"L'interm6diaire des math., 20, 1913, 202-6. 
