168 HISTORY OF THE THEORY OP NUMBERS. [CHAP, vi
*Th. Schroder66 and J. Hartmann57 treated periodic decimals.
W, Shanks58 gave Lambert's method (Bernoulli,8 end) for shortening the work of finding the length of the period for 1/N.
G. Salmon59 remarked that the number n of digits in the period is known if we find two remainders which are powers of 2, since 10a^=2p and 106ss'2fl imply 10afl~6l>s= 1; also if we find three remainders which are products of powers of 2 and 3. Muir71 noted that it is here implied that aq—bp equals n, whereas it is merely a multiple of n.
J. W. L. Glaisher60 proved that, for any base r,
= .012...r-3r-l,
a generalization of 1/81 = .012345679.
W. Shanks61 gave the length of the period for 1/p, when p is a prime < 30000, and a list of 69 errors or misprints in the table by Desmarest,88 and 11 in that by Burckhardt.20
Shanks62 gave primes p for which the length n of the period for 1/p is a given number ^100, naturally incomplete. Shanks63 gave additional entries p for n=26, n = 99; noted corrections to his former table and stated that he had extended the table to 40000. Shanks64 mentioned an extension in manuscript from 40000 to 60000. An extension to 120000 in manuscript was made by Shanks, 1875-1880. The manuscript, described by Cunningham,124 who gave a list of errata, is in the Archives of the Royal Society of London.
Shanks65 stated that if a is the length of the period for 1/p, where p is a prime >5, that for l/pn is ap""1 {without the restriction by Thibault,31 Muir71].
G. de Coninck66 stated that, if the last digit (at the right) of A is 1 or 9, the last digit of the period for I/A is 9 or 1; while, if A is a prime not ending in 1 or 9, its last digit is the same as the last in the period.
Moret-Blanc67 noted that the last property holds for any A not divisible by 2 or 5. For, if a is the integer defined by the period for I/A, that for (A —I)/A is (A —l)a, whence a+(A — l)a = 10n—1, if n is the length of the periods. He noted corrections to the remaining nine laws stated by Coninck and implied that when corrected they become trivial or else known facts.
"Progr. Ansbach, 1872.
"Progr. Rinteln, 1872.
"Messenger Math., 2, 1873, 41-43.
"/bid., pp. 49-51, 80.
M/Wd., p. 188.
•lProc. Roy. Soc. London, 22, 1873-4, 200-10, 384-8. Corrections by Workman.117