Chap. VI] PERIODIC DECIMAL FRACTIONS. 173 
p — l=sh and we have h sets of s fractions whose periods differ only by the 
cycHc permutation of the digits. If p is a product of distinct primes pi, P2 • • • 
and if the lengths of the periods of 1/p, 1/pi, l/p2, ■ ■ ■ are s, Si, S2, . . . , then 
s is the 1. c. m. of Si, S2,.... If p = Pi"P2^- • •> and s, Si, s' are the lengths of 
the periods of 1/p, 1/pi, l/pi", then s' is one of the numbers Si, Sip,. . ., 
SiPi°~^ and hence divides (pi — l)pi"~^; and s is a divisor of <f){p). Thus p 
divides lO^^^^-l. 
C. A. Laisant^^ used a lattice of points, whose abscissas are a+r,a-\-2r,..., 
a-\-pf and ordinates are their residues <p modulo p, to represent graphically 
periodic decimal fractions and to expand fractions into a difference of two 
series of ascending powers of fixed fractions. 
*A. Rieke^® noted that a periodic decimal with a period of 2m digits equals 
(i4. + l)/(10"*+l), where A is the first half of the period. He discussed the 
period length for any base. 
W. E. HeaP^ noted that, if B contains all the prime factors of N, the 
number of digits in the fraction to the base B for M/N is the greatest integer 
in (n+n' — l)/n', where n—n' is the greatest difference found by subtracting 
the exponent of each prime factor of N from the exponent of the same prime 
factor of B. If B contains no prime factor of N, the fraction for M/N is 
purely periodic, with a period of ^(A'') digits. If B contains some, but not 
all, of the prime factors of N, the number of digits preceding the period is 
the same as in the first theorem. The proofs are obscure. There is given 
the period for 1/p when p<100 and has 10 as a primitive root [the same 
p's as by Glaisher^^]. Likewise for base 12, with p<50. 
R. W. Genese^^ noted that, if we multiply the period for 1/81 [Glaisher^"] 
by m, where m<81 and prime to it, we get a period containing the digits 
0, 1, . . ., 9 except 9n—m, where 9n is the multiple of 9 just exceeding m. 
Jos. Mayer^^ investigated the moduli with respect to which 10 belongs 
to a given exponent, and gave the factors of 10"— 1, n< 12. He discussed 
the determination of the exponent to which 10 belongs for a given modulus 
by use of the theory of indices and by the methods of quadratic, cubic, 
biquadratic,... residues. He used also the fact that there are (a — a') 
08-/3') . . . divisors of Pi''p2^Vz ■ ■ • which divide no one of the fixed factors 
ViVVi ■ • ■, Pi>2W> • • • J where a<a,b<^,..., and pi, P2,-- are distinct 
primes. He gave the length of the period for 1/p, for each prime p^2543 
and 22 higher primes [Burckhardt^^]. 
L. Contejean^°° proved that, in the conversion of an irreducible fraction 
a/h into a decimal fraction, if the remainders o^ and a^ are congruent 
modulo b, so that lO'a^lO^'a, then 10"'~''-1 is divisible by the quotient 
h' of b by the highest factor 2*5' of b. Thus the length of the period is 
"Assoc, fran?. avanc. sc, 16, 1887, II, 228-235. 
•'Versuch iiber die periodischen Bniche, Progr., Riga, 1887. 
•^Annals of Math., 3, 1887, 97-103. 
•^Report Britiah Assoc, 1888, 580-1. 
"Ueber die Grosse der Periode eines unendlichen Dezimalbruches, oder die Congruence 
lO^Sl (mod P). Progr. K. Studienanstalt Burghausen, Munchen, 1888, 52 pp. 
""Bull. Boc. philomathique de Paris, (8), 4, 1891-2, 64-70. 
