172 History of the Theory of Numbers. [Chap, vi 
of the period. The property of Schlomilch holds for these and the generali- 
zation to any base, as well as for those with the law xjm = '^ym-\-\-ym-2' But if 
ym = ^ym-i-2y^.2 (mod 9), ?/. = (2"'-'-l)(!/2-2/i)+2/i (mod 9), 
the fact that 2^ = 1 (mod 9) shows that the period has at most six digits. 
Those with six reduce by cychc permutation to nine periods : 
167943, 235986, 278154, 197346, 
265389, 218457, 764913, 329568, 751248. 
In the A-th of these the sum of corresponding digits in the two half periods 
is always =A- (mod 9). 
Karl Broda^^ examined for small values of r and certain primes p the 
solutions a: of x''= 1 (mod p) to obtain a base x for which the periodic frac- 
tion for 1/p has a period of r digits, and similariy the condition x^=—\ 
(mod p) for an even number of digits in the period (Broda^^). 
F. Kessler^^ factored 10"-1 forn = ll, 20, 22, 30. 
W. W. Johnson^° formed the period for 1/19 by placing 1 at the extreme 
right, next its double, etc., marking wdth a star a digit when there is 1 to carry: 
« * * * * * * «« 
05263157894736842 1. ;^ 
To deduce the value of 1/19 written to the base 2, use 1 for each digit 
starred and for the others, reversing the order: 
.6 0001101011110010 i. 
If we apply the first process with the multipUer m, we get the period for the 
reciprocal of 10?7i — 1. 
E. Lucas^^ gave the prime factors of 10" — 1 for n odd, n^l7, 7i = 21, 
and certain factors forn = 19, . . . , 41 ; those of 10" + 1 for n^ 18 and n = 21. 
He stated that the majority of the results were given by Loof and pubUshed 
by Reuschle. In 1886, Le Lasseur gave 
10^7-1 =3--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. ^-^ 
F. Kessler^- listed nine errors in Burckhardt's-" table and described his 
own manuscript of a table to p = 12553, i. e., for the first 1500 primes. 
Van den Broeck^^ stated that 10^" -1 is divisible by 3"+^ 
A. Lugli^^ proved that, if p is a prime 5^2, 5, the length of the period 
of 1/p is a divisor of p — 1. If the number of digits in the period of a/p is 
an even number 2t, the ^th remainder on dividing a by p is p — 1, and con- 
versely. Hence, if r^ is the hth remainder, rh+rh+i = p {h = l,. . ., t), and 
the sum of all the r's is tp. If the period of 1/p has s digits, s<p — 1, then 
".\rchiv Math. Phys., 68, 1882, 85-99. 
«»Zeitschrift Math. Naturw. Unterricht, 15, 1884, 29. 
•"Messenger of Math., 14, 1884-5, 14-18. 
"Jour, de math. 6Um., (2), 10, 1886, 160. Cf . rinterm^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. 
"Mathesis, 6, 1886, 70. Proofs, 23.5-6, and Math. Quest. Educ. Times, 54, 1891, 117. 
"Periodico di Mat., 2, 1887, 161-174. 
