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```CHAP, vi]                    PERIODIC DECIMAL FRACTIONS.                           171
place of the product ap, /3 the digit in the tens place. Write the digit b to the left of digit a to form the last two digits of the required number P. The number c in the units place in 6p+/3 is written to the left of digit b in P. To cp add the digit in the tens place of bp and place the unit digit of the sum to the left of c in P. The process stops with the kth digit t if the next digit would give a. Then P = t . . . cba and its products by k integers or fractions has the same k digits in the same cyclic order. For a = 2, p = 3, we get /c = 28 and see that P is the period of 2/27, and the k multipliers are w/2, m = l, . . ., 28. [To have an example simpler than the author's, take o=7, p = 5; then P — 142857, the period of 1/7; the multipliers are 1,. . ., 6.] For proof, we have
^+ . . . +102c+106+a,        PP = lO^a+lO*-2^ . . . +10c+b,
a   =     ** IQp-l    10*-!'
so that P is the period with k digits for a/(10p — 1).
E. Lucas82 gave the prime factors of 1018*1, 1017±1, 1021±1, 1016+1, 1018+1, communicated to him by W. Loof, with the remark that (1019- 1)/9 has no prime factor < 3035479. Lucas gave the factors of 1030+1.
J. W. L. Glaisher83 proved his76 earlier statements, repeated his77 earlier remarks, and noted that, if q is a prune such that the period for 1/q has q — 1 digits, the products of the period for 1/q by 1, 2, . . ., q — 1 have the same digits in the same cyclic order. This property, well known for q = 7, holds also for g = 17, 19, 23, 29, 47, 59, 61, 97 and for g = 72.
0. Schlomilch84 stated that, to find every N for which the period for l/N has 2k digits such that the sum of the sth and (&+s)th digits is 9 for 5 = 1, . . ., k, we must take an integer N= (10*+1)/!T; then the first k digits of the period are the k digits of T— 1.
C. A. Laisant85 extended his investigations with Beaujeux52'66 and gave a summary of known properties of periodic fractions; also his86 process to find the period of simple periodic fractions without making divisions.
V. Bouniakowsky87 noted that the property of the period of 1 A". observed by Schlomilch84 for N = 7, 11, 13, 77, 91, 143, holds also for the periods of k/N, for fc = Ar— land (N~l)/2, with the same values of AT". Consider the decimal fraction Q.yiy2 • . . with ym=ym-i+ym-2, (mod 9), replacing any residue zero by 9, and taking ?/i>0, 2/2>0. The fraction is purely periodic and is either 0.9 or 0.33696639 or has the same digits permuted cyclically, or else has a period of 24 digits and begins with 1, 1 or 2, 2 or 4, 4, or has the same 24 digits permuted cyclically or by the interchange of the two halve?
82Nouv. Corresp. Math., 5, 1879, 138-9.
83Nature, 19, 1879, 208-9.
"Zeitschrift Math. Phye., 25, 1880, 416.
«M6m. Soc. Sc. Phys. et Nat. de Bordeaux, (2), 3, 1880, 213-34.
MLes Mondes, 19, 1869, 331.
"Bull. Acad. Sc. St*P<§tersbourg, 27, 1881, 362-9.```