Chap. VI] PERIODIC DECIMAL FRACTIONS. 161 
Anton FelkeP^ showed how to convert directly a periodic fraction 
written to one base into one to another base. He gave all primes < 1000 
which can divide a period with a prime number of digits <30, as 29m +1 
= 59,233,.... 
Oberreit^* extended Bernoulli's^ table of factors of 10*=*=!. 
C. F. Gauss^^ gave a table showing the period of the decimal fraction for 
Vp", p"<467, V a prime, and the period for 1/p", 467^ p"^ 997. 
W. F. Wucherer^® gave five places of the decimal fraction for n/d, 
d<1000, n<dford<50, n^ 10 for (^^ 50. 
Schroter published at Helmstadt in 1799 a table for converting ordinary 
fractions into decimal fractions. 
C. F. Gauss^'^ proved that, if a is not divisible by the prime p (^7^2, 5), 
the length of the period for a/p" is the exponent e to which 10 belongs 
modulo p^. If we set 0(p") =ef and choose a primitive root r of p^ such 
that the index of 10 is /, we can easily deduce from the periods for k/p^, 
where k = \, r, . . ., r^~\ the period for m/p", where m is any integer not 
divisible by p. For, if i be the index of m to the base r, and if i = af-\-^, 
where 0^/3</, we obtain the period for m/p" from that for rVp" by carrying 
the first a digits to the end. He computed^^ the necessary periods for each 
p"<1000, but published here the table only to 100. By using partial 
fractions, we may employ the table to obtain the period for a/b, where b 
is a product of powers of primes within the limits of the table. 
H. Goodwyn^^ noted that, if a<17, the period for a/17 is derived from 
the period for 1/17 by a cyclic permutation of the digits. Thus we may 
print in a double line the periods for 1/17, . . . , 16/17 by showing the period 
for 1/17 and, above each digit d of the latter, showing the value of a such 
that the period for a/ 17 begins with the digit d, while the rest of the 
period is to be read cyclically from that for 1/17. 
Goodwyn^^ noted that when 1/p is converted into a decimal fraction, 
p being prime, the sum of corresponding quotients in the two half periods 
is 9, and that for remainders is p, if p^7. 
J. C. Burckhardt^" gave the length of the period for 1/p for each prime 
p^2543 and for 22 higher primes. It follows that 10 is a primitive root 
of 148 of the 365 primes p, 5<p<2500. 
"Abhand. Bohmiachen Gesell. Wias., Prag, 1, 1785, 135-174. 
"J. H. Lambert's Deutscher Gelehrter Briefwechsel, pub. by J. Bernoulli, Leipzig, vol. 5, 
1787, 480-1. The part (464-479) relating to periodic decimals is mainly from Ber- 
noulli's' paper. 
"Posthumous manuscript, dated Oct., 1795; Werke, 2, 1863, 412-434. 
^'Beytrage zum allgemeinem Gebrauch der Decimal Brliche. . . ., Carlsruhe, 1796. 
"Disq. Arith., 1801, Arts. 312-8. A part was reproduced by Wertheim, Elemente der Zahlen- 
theorie, 1887, 153-6. 
I'Jour. Nat. Phil. Chem. Arts (ed., Nicholson), London, 4, 1801, 402-3. 
"76id., new series, 1, 1802, 314-6. Cf. R. Law, Ladies' Diary, 1824, 44-45, Quest. 1418. 
'"Tables des diviseurs pour tous les nombres du premier milMon, Paris, 1817, p. 114. For errata 
see Shanks," Kessler," Cimningham,^^! and G^rardin."^ 
