Chap. VI] PeEIODIC DECIMAL FRACTIONS. 165 
Next, if D = 10k — 1, we have a like rule to be applied only to the ^"^ — 1. If 
D = 10k=^3, 1/(3Z)) has a denominator lOZ^l, and the length of its period, 
found as above, is shown to be not less than that for 1/D. 
Th. Bertram^^ gave certain numbers p for which l/p has a given 
length k of period for k^ 100. Cf. Shanks.^^ 
J. R. Young^^ took a part of a periodic decimal, as .1428571 428 for 1/7, 
and marked off from the end a certain number (three) of digits. We can 
find a multipHer (as 6) such that the product, with the proper carrying 
(here 2) from the part marked off, has all the digits of the abridged number 
in the same cyclic order, except certain of the leading digits. In the special 
case the product is .8571428. 
W. Loof" gave the primes p for which the period for l/p has a given 
number n of digits, n^ 60, with no entry for n = 17, 19, 37-40, 47, 49, 57, 59, 
and with doubt as to the primality of large numbers entered for various 
other n's. 
E. Desmarest^^ gave the primes P< 10000 for which 10 belongs to the 
exponent {P — l)/t for successive values of t. The table thus gives the 
length of the period for 1/P. He stated (pp. 294-5) that if P is a prime 
< 1000, and if p is the length of the period for A/P, then except for P = 3 
and P = 487 the length of the period for A/P^ is pP. 
A. Genocchi^^ proved Euler's^^ rule by use of the quadratic reciprocity 
law. Thus 5 is a quadratic residue or non-residue of N according as 
N = 5m=^l or 5m±3; for 4n+l = 5m='=l, n or n — 2 is divisible by 5; for 
4n — l = 5m='=l, n or n-\-2 is divisible by 5. Also, 2 is a residue of 4n±l 
for n even, a non-residue for n odd. Hence 10 is a residue of A^ = 4n='= 1 for 
n even if n orn =f2 is divisible by 5, and for n odd if neither is. Thus Euler's 
inclusion of n=F6 is superfluous. By a similar proof, 10 is quadratic non- 
residue of A/' = 4n±l if both 2 and 5 occur among the divisors of n±2, 
n±6, or if neither occurs; a residue if a single one of them occurs. 
A. P. Reyer^^" noted that the period for a/3^ has 3^~^ digits and gave the 
length of the period for a/p for each prime p< 150. 
*F. van Henekeler^^^ treated decimal fractions. 
C. G. Reuschle^" gave for each prime p< 15000 the exponent e to which 
10 belongs modulo p. Thus e is the length of the period for l/p. He gave 
all the prime factors of lO'^-l for n^l6, n = lS, 20, 21, 22, 24, 26, 28, 30, 
32, 36, 42; those of 10"+1 for n^l8, n = 21; also cases up to n = 243 of 
the factors of the quotient obtained by excluding analytic factors. 
"Einige Satze aus der Zahlenlehxe, Progr. Coin, Berlin, 1849, 14-15. 
»«London, Ed. Dublin Phil. Mag., 36, 1850, 15-20. 
»^Archiv Math. Phys., 16, 1851, 54-57. French transl. in Nouv. Ann. Math., 14, 1855, 115-7. 
Quoted by Brocard, Mathesis, 4, 1884, 38. 
'^Th^orie des nombres, Paris, 1852, 308. For errata, see Shanks*^ and G^rardin.^'^ 
"Bull. Acad. Roy. Sc. Belgique, 20, II, 1853, 397-400. 
"<^Archiv Math. Phys., 25, 1855, 190-6. 
'^''Ueber die primitiven Wurzeln der Zahlen und ihre Anwendung auf Dezimalbriicbe, Leyden, 
1855 (Dutch). 
"Math. Abhandlung.. .Tabellen, Progr. Stuttgart, 1856. Full title in Ch. I."* Errata, 
Bork,i''5 Hertzer,ii3 Cunningham. i" 
I 
