CHAP, vi] PERIODIC DECIMAL FRACTIONS. 165 Next, if D = 10&—1, we have a like rule to be applied only to the km—1. If D = 10&=fc 3, I/(3D) has a denominator 10Z=t 1, and the length of its period, found as above, is shown to be not less than that for 1/D. Th. Bertram36 gave certain numbers p for which 1/p has a given length k of period for k^ 100. Cf. Shanks.62 J. R. Young36 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 multiplier (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. Loof37 gave the primes p for which the period for 1/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. Desmarest38 gave the primes P< 10000 for which 10 belongs to the exponent (P—!)/£ 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/P2 is pP. A. Genocchi39 proved Euler's11 rule by use of the quadratic reciprocity law. Thus 5 is a quadratic residue or non-residue of N according as JV=5m=t:l or Sw^S; for 4n+las5wd=l, n or n—2 is divisible by 5; for 4n—l = 5m=fcl, 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 N=4/1=*= 1 for n even if n or n =p2 is divisible-by 5, and for n odd if neither is. Thus Euler's inclusion of n=p6 is superfluous. By a similar proof, 10 is quadratic non-residue of -/V = 4n=fcl 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. Reyer390 noted that the period for a/3p has 3P~2 digits and gave the length of the period for a/p for each prime p< 150. *F. van Henekeler39& treated decimal fractions. C. G. Reuschle40 gave for each prime p< 15000 the exponent e to which 10 belongs modulo p. Thus e is the length of the period for 1/p. He gave all the prime factors of 10n-l for ng!6, n = 18, 20, 21, 22, 24, 26, 28, 30, 32, 36, 42; those of 10n+l for n^!8, n = 21; also cases up to n = 243 of the factors of the quotient obtained by excluding analytic factors. «Einige Satze aus der Zahlenlehre, Progr. Coin, Berlin, 1849, 14-15. '•London, Ed. Dublin Phil. Mag., 36, 1850, 15-20. «7Archiv Math. Phys., 16, 1851, 54-57. French transl. in Nouv. Ann. Math., 14, 1855, 115-7. Quoted by Brocard, Mathesis, 4, 1884, 38. a8The"orie des nombres, Paris, 1852, 308. For errata, see Shanks81 and Ge>ardin.131 "Bull. Acad. Roy. Sc. Belgique, 20, II, 1853, 397-400. »aArchiv Math. Phys., 25, 1855, 190-6. "bUeber die primitiven Wurzeln der Zahlen und ihre Anwendung auf Dezimalbrucbe, Leyden, 1855 (Dutch). <°Math. Abhandlung.. .Tabellen, Progr. Stuttgart, 1856. Full title in Ch. I.108 Errata, Bork,105 Hertzer,"' Cunningham.121