172 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vi of the period. The property of Schlomilch holds for these and the generalization to any base, as well as for those with the law ymzz2ym-i+ym-2. But if ym=3ym-i~2ym-2 (mod 9), 2/ro-(27n-1-l)(2/2-2/i)+2/i (mod 9), the fact that 26=1 (mod 9) shows that the period has at most six digits. Those with six reduce by cyclic permutation to nine periods: 167943, 235986, 278154, 197346, 265389, 218457, 764913, 329568, 751248. In the kth of these the sum of corresponding digits in the two half periods is always =k (mod 9). Karl Broda88 examined for small values of r and certain primes p the solutions x of xr=l (mod p) to obtain a base x for which the periodic fraction for l/p has a period of r digits, and siinilarly the condition zrs= -1 (mod p) for an even number of digits in the period (Broda68). F. Kessler89 factored 10ft-l for n = ll, 20, 22, 30. W. W. Johnson90 formed the period for 1/19 by placing 1 at the extreme right, next its double, etc., marking with a star a digit when there is 1 to carry: 052631578947368421. To deduce the value of 1/19 written to the base 2^ use 1 for each digit starred and 0 for the others, reversing the order: .6 0001101011110010 1. If we apply the first process with the multiplier m, we get the period for the reciprocal of 10m — 1. E. Lucas91 gave the prune factors of 10n—1 for n odd, n^!7, n = 21, and certain factors for n = 19,..., 41; those of 10n+l for rc^ 18 and n = 21. He stated that the majority of the results were given by Loof and published by Reuschle. In 1886, Le Lasseur gave 1017-1 = 32-2071723-5363222[3]57, said by Loof to have no divisor < 400,000 other than 3,9. On the omission of the digit 3, see Cunningham.123 F. Kessler92 listed nine errors in Burckhardt's20 table and described his own manuscript of a table to p = 12553, i. e., for the first 1500 primes. Van den Broeck93 stated that 103"-1 is divisible by 3n+2. A. Lugli94 proved that, if p is a prime 7*2, 5, the length of the period of l/p is a divisor of p — 1. If the number of digits in the period of a/p is an even number 2t, the tth remainder on dividing a by p is p — l, and conversely. Hence, if rh is the /ith remainder, rh+rh+t = P (h = l,..., t), and the sum of all the r's is tp. If the period of l/p has s digits, s<p-1, then "Archiv Math. Phys., 68,1882, 85-99. "Zeitschrift Math. Naturw. Unterricht, 15, 1884, 29. •"Messenger of Math., 14, 1884-5, 14-18. "Jour, demath. e!6m., (2), 10,1886,160. Cf. 1'interme'diaire des math., 10,1903,183. Quoted by Brocard, Mathesis, 6, 1886, 153; 7,1887, 73 (correction, 1889, 110). "Archiv Math. Phys., (2), 3,1886, 99-102. 93Mathesis, 6, 1886, 70. Proofs, 235-6, and Math. Quest. Educ. Times, 54, 1891, 117 "Periodico di Mat., 2,1887,161-174.