Chap. VI] PERIODIC DECIMAL FRACTIONS. 167 
V. A. Lebesgue^^ gave for iV^347 the periods for 1/iV, r/N,. . .[cf. 
Gauss^l. 
Sanio^° stated that, if m, n,. . . are distinct primes and 1/m, 1/n, . . . 
have periods of length q, q',. . ., then l/(wV. . .) has the period length 
^a-i^b-i qq' ^ He gave the length of the period for l/p for each 
prime p^700, and the factors of 10" — 1, n^ 18. 
F. J. E. Lionnet^^ stated that, if the period for a/h has n digits, that for 
any irreducible fraction whose denominator is a multiple of h has a multiple 
of n digits. If the periods for the irreducible fractions a/6, a'/h', . . . have 
n, n', . • • digits, every irreducible fraction whose denominator is the 1. c. m. 
of b, h',. . . has a period whose length is the 1. c. m. of n,n',. . .. If the period 
for 1/p has n digits and if p" is the highest power of the prime p which divides 
10" — 1, any irreducible fraction with the denominator p"'^^ has a period 
of np^ digits. 
C. A. Laisant and E. Beaujeux^^ proved that if g is a prime and the 
period for 1/q to the base B is P = ab. . .h, with q — 1 digits, then 
P-{a+h+...+h) = {B-l)a, ^{^+^-y) = 
B'-'-l 
and stated that a like result holds for a composite number q if we replace 
q—1 by/=</)(g). Their proof of the generaUzed Fermat theorem 5^=1 
(mod q) is quoted under that topic. 
C. Sardi^^ noted that if 10 is a primitive root of a prime p = lOn+1, the 
period for 1/p contains each digit 0,..., 9 exactly n times [Hudson^^]. 
For p = 10n-f-3, this is true of the digits other than 3 and 6, which occur 
n+1 times. Analogous results are given for lOn+7, lOn+9. 
Ferdinand Meyer^^ proved an immediate generalization from 10 to 
any base k prime to 6, 6', ... of the statements by Lionnet.^^ 
Lehmann^" gave a clear exposition of the theory. 
C. A. Laisant and E. Beaujeux^^ considered the residues Vq, ri, . . . when 
A, AB, AB^,. . . are divided by A- Let ri_iB = QiZ)i+r,. When written 
to the base B, let Di = ap. . .02^1, and set Di = ap. . Mi. Then 
airi+ . . . +0prp = Z)i(ri-Q2A- • • • -QpDp). 
The further results are either evident or not novel. 
For G. Barillari^°" on the length of the period, see Ch. VII. 
*'M6m. soc. sc. phys. et nat. de Bordeaux, 3, 1864, 245. 
^"Ueber die periodischen Decimalbrtiche, Progr., Memel, 1866. 
"Algebre 61em., ed. 3, 1868. Nouv. Ann. Math., (2), 7, 1868, 239. Proofs by Morel and Pellet, 
(2), 10, 1871, 39-42, 92-95. 
MNouv. Ann. Math., (2), 7, 1868, 289-304. 
"Giornale di Mat., 7, 1869, 24-27. 
"Archiv Math. Phys., 49, 1869, 168-178. 
""Ueber Dezimalbriiche, welche aus gewohnUchen Briichen abgeleitet sind, Progr., Leipzig, 1869. 
66N0UV. Ann. Math., (2), 9, 1870, 221-9, 271-281, 302-7, 354-360. 
