CHAP, vi] PEBIODIC DECIMAL FRACTIONS. 161 Anton Felkel13 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,.... Oberreit14 extended Bernoulli's9 table of factors of 10*=*= 1. C. F. Gauss15 gave a table showing the period of the decimal fraction for k/pn, pn<467, p a prime, and the period for l/pn, 467^pn^997. W. F. Wucherer16 gave five places of the decimal fraction for n/dt d<1000, n<d for d<50, nglO for <^50. Schroter published at Helmstadt in 1799 a table for converting ordinary fractions into decimal fractions. C. F. Gauss17 proved that, if a is not divisible by the prime p (p?*2, 5), the length of the period for a/pn is the exponent e to which 10 belongs modulo pn. If we set <t>(pn) =ef and choose a primitive root r of pn such that the index of 10 is /, we can easily deduce from the periods for k/pn, where fc = l, r,..., r7""1, the period for m/pn, where m is any integer not divisible by p. For, if i be the index of m to the base r, and if £ = a/+/3, where 0^/3</, we obtain the period for m/pn from that for r^/pn by carrying the first a digits to the end. He computed15 the necessary periods for each pn<1000, but published here the table only to 100. By using partial fractions, we may employ the table to obtain the period for a/6, where 6 is a product of powers of primes within the limits of the table. H. Goodwyn18 noted that, it 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. Goodwyn19 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. Burckhardt20 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. Bahmischen Gesell. Wise., Prag, 1, 1785, 135-174. 14J. 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 Bernoulli's9 paper. "Posthumous manuscript, dated Oct., 1795; Werke, 2, 1863, 412-434. "Beytrage zum allgemeinern Gebrauch der Decimal Briiche...., Carlsruhe, 1796. 17Disq. Arith., 1801, Arts. 312-8. A part was reproduced by Wertheim, Elemente der Zahlen-theorie, 1887, 153-6. "Jour. Nat. Phil. Chera. Arts (ed., Nicholson), London, 4, 1801, 402-3. »/6id., new series, 1, 1802, 314-6. Cf. R. Law, Ladies' Diary, 1824, 44r45, Quest. 1418. a°Tables des diviseurs pour tous les nombres du premier million, Paris, 1817, p. 114. For errata see Shanks," Keasler,83 Cunningham,114 and Ge'rardm."1