History Of The Theory Of Numbers - I

CHAP, vi] PERIODIC DECIMAL FRACTIONS. 167
V. A. Lebesgue49 gave for N^347 the periods for 1/N, r/N,. ..[cf. Gauss17].
Sanio50 stated that, if w, n, . . . are distinct primes and 1/w, 1/rc, . . . have periods of length q, q', . . ., then l/(manb . . .) has the period length m0"1?!6"1. . .qq' . . .. He gave the length of the period for 1/p for each prime p^TOO, and the factors of 10n-l, n£l&.
F, J. E. Lionnet61 stated that, if the period for a/b has n digits, that for any irreducible fraction whose denominator is a multiple of b has a multiple of n digits. If the periods for the irreducible fractions a/b, a'/b', . . . have n, n', . . . digits, every irreducible fraction whose denominator is the 1. c. m. of b, b'y . . . has a period whose length is the 1. c. m. of n, n', . . . . If the period for l/p has n digits and if pa is the highest power of the prime p which divides 10n — 1, any irreducible fraction with the denominator pa+fi has a period of np0 digits.
C. A. Laisant and E. Beaujeux62 proved that if q is a prime and the period for 1/q to the base B is P = ab. . .h, with q~l digits, then
and stated that a like result holds for a composite number q if we replace q — l by /=<£(<?)• Their proof of the generalized Fermat theorem jB7=l (mod q) is quoted under that topic.
C. Sardi63 noted that if 10 is a primitive root of a prime p = 10n+l, the period for l/p contains each digit 0,..., 9 exactly n times [Hudson48]. For p = 10n+3, this is true of the digits other than 3 and 6, which occur n-fl times. Analogous results are given for Wn+7, 10n+9.
Ferdinand Meyer54 proved an immediate generalization from 10 to any base k prime to 6, b', . . . of the statements by Lionnet.51
Lehmann640 gave a clear exposition of the theory.
C. A. Laisant and E. Beaujeux65 considered the residues r0, r1} . . . when A, AB, A£2,. . . are divided by JDj. Let rt-_1B = QiD1+^ When written to the base B, let Di = ap. . . a2a1} and set Di = ap. . . a». Then
OiriH- • . . +aprp=D1(r1-Q2D2- . . . -QPDP).
The further results are either evident or not novel.
For G. Barillari60a on the length of the period, see Ch. VII.
49M6m. soc. sc. phys. et nat. de Bordeaux, 3, 1864, 245.
BOUeber die periodischen Decimalbrttche, Progr., Memel, 1866.
"Alg&bre 616m., ed. 3, 1868. Nouv. Ann. Math., (2), 7, 1868, 239. Proofs by Morel and Pellet,
(2), 10, 1871, 39-42, 92-95. "Nouv. Ann. Math., (2), 7, 1868, 289-304. B3Giornale di Mat., 7, 1869, 24-27. O'Archiv Math. Phys., 49, 1869, 168-178.
64flUeber Dezimalbriiche, welche aus gewohnlichen Brtichen abgeleitet sind, Progr., Leipzig, 1869. 66Nouv. Ann. Math., (2),. 9, 1870, 221-9, 271-281, 302-7, 354-360.