Chap. VI] PERIODIC DECIMAL FRACTIONS. 175 
to N, and D is a multiple of the exponent d to which N belongs modulo P. 
The further discussion is limited to the case D = d, to exclude repetitions 
of the period of digits. Then the multipUers which cause a cyclic permuta- 
tion of the digits are the least residues of N, N^, . . . , A^^ modulo P. For 
A = 1, we have a solution for any N and any P prime to N. There are listed 
the 19 possible solutions with A>1, N^QS, and having the first digit >0. 
The only one with A^= 10 is 142857. General properties are noted. 
A like form is obtained (pp. 375-7) for an integer of D digits written to 
the base A^, such that its quotients by D distinct integers have the same 
D digits in the same cyclic order. The divisors are the least residues of 
N^, N^-\. . ., N modulo P. For example, if N = n, P = 7, A=4:, we get 
4(11^ — 1)/7, or 631 to base 11, whose quotients by 2 and 4 are 316 and 163, 
to base 11. Another example is 512 to base 9. 
E. Lucas^ gave all the prime factors of 10"— 1 forn^ 18. 
F. W. Lawrence^"^ proved that the large factors of 10^^ — 1 and 10^^ — 1 
are primes. 
C. E. Bickmore^'^^ gave the factors of 10" -1, n^ 100. Here (1023-l)/9 
is marked prime on the authority of Loof , whereas the latter regarded its 
composition as unknown [Cunningham^^^]. There is a misprint for 43037 
in 10^^-1. 
B. Bettini^"^ considered the number n of digits in the period of the deci- 
mal fraction for a/b, i. e., the exponent to which 10 belongs modulo h. If 
10 is a quadratic non-residue of a prime b, n is even, but not conversely 
(p. 48). There is a table of values of n for each prime 6^277. 
V. Murer^^" considered the n = mq remainders obtained when a/b is 
converted into a decimal fraction with a period of length n, separated them 
into sets of m, starting with a given remainder, and proved that the sum 
of the sets is a multiple of 9 ... 9 (to m digits) . Further theorems are found 
when q = l, 2 or 3. 
J. Sachs ^^^^ tabulated all proper fractions with denominators <250 and 
their decimal equivalents. 
B. Reynolds^ ^^ repeated the rules given by Glaisher'^^' '^^ for the length 
of periods. He extended the rules by Sardi^^ and gave the number of times 
a given digit occurs in the various periods belonging to a denominator N, 
both for base 10 and other bases. 
Reynolds^^^ gave numerical results on periodic fractions for various 
bases the lengths of whose period is 3 or 6, and on the length of the period for 
1/A^ for every base <N—1, when A^ is a prime. 
A. Cunningham^ ^^ applied to the question of the length of the period 
of a periodic fraction to any base the theory of binomial congruences [see 
i"Proc. London Math. Soc, 28, 1896-7, 465. Ci. Bickmore" of Ch. XVI. 
"'Nouv. Ann. Math., (3), 15, 1896, 222-7. 
"Teriodico di Mat., 12, 1897, 43-50. "o/bid., 142-150. 
uoaprogr. 632, Baden-Baden, Leipzig, 1898. 
""Messenger Math., 27, 1897-8, 177-87. 
»"/feid., 28, 1898-9, 33-36, 88-91. 
"'/bid., 29, 1899-1900, 145-179. Errata.i" 
