CHAP, vi] PEEIODIC DECIMAL FBACTIONS. 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 Z) = d, to exclude repetitions of the period of digits. Then the multipliers which cause a cyclic permutation of the digits are the least residues of AT, N2,..., ND 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, JV^63, and having the first digit >0. The only one with N = 10 is 142857. General properties are noted.
A like form is obtained (pp. 375-7) for an integer of D digits written to the hase N, 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 ND, ND~\..., N modulo P. For example, if N*= 11, P=7, A = 4, we get 4(ll3-l)/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. Lucas1 gave all the prune factors of 10n—1 for n^ 18.
F. W. Lawrence107 proved that the large factors of 1025-1 and 1029-1 are prunes.
C. E. Bickmore108 gave the factors of 10n-1, ng 100. Here (1023 -1)/9 is marked prime on the authority of Loof, whereas the latter regarded its composition as unknown [Cunningham123]. There is a misprint for 43037 in 1029-1.
B. Bettini109 considered the number n of digits in the period of the decimal fraction for afb, i. e., the exponent to which 10 belongs modulo b. If 10 is a quadratic non-residue of a prime 6, n is even, but not conversely (p. 48). There is a table of values of n for each prime &^277.
V. Murer110 considered the n = mq remainders obtained when a/6 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. Sachs1100 tabulated all proper fractions with denominators < 250 and their decimal equivalents.
B. Reynolds111 repeated the rules given by Glaisher78'79 for the length of periods. He extended the rules by Sardi53 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.
Reynolds112 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 l/N for every base <N — 1, when N is a prime.
A. Cunningham113 applied to the question of the length of the period of a periodic fraction to any base the theory of binomial congruences [see
107Proc. London Math. Soc., 28, 1896-7, 465. Cf. Bickmore*9 of Ch. XVI.
l08Nouv. Ann. Math., (3), 15, 1896, 222-7.
i°9Periodico di Mat., 12, 1897, 43-50. "'/bid., 142-150
uoaProgr. 632, Baden-Baden, Leipzig, 1898.
inMessenger Math., 27, 1897-8, 177-87.
112/6id., 28, 1898-9, 33-36, 88-91.
»3jm, 29, 1899-1900, 145-179. Errata.118