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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:
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 10n1 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.