164 HISTORY OF THE THEORY OP NUMBERS. [CHAP, vi
E. Catalan30 converted periodic decimals into ordinary fractions without using infinite progressions. When 1/13 is converted into a decimal, the period of remainders is 1, 10, 9, 12, 3, 4; repeat the period; starting in the series of 12 terms with any term (as 10), take the fourth term (4) after it, the fourth term (12) after that, etc.; then the sum 26 of the three is a multiple of 13. In general, if D is a prime and D — l=wn, the sum of n terms taken m by m in the period for N/D is a multiple of D [cf . Thibault31].
If the sum of two terms of the period of remainders for N/D is D, the same is true of the terms following them. Hence the sum of corresponding terms of the two half periods is D. This happens if the number of terms of the period is 0(Z>).
Thibault31 denoted the numbers of digits hi the periods for 1/d and 1/d' by m and m'. If d' is divisible by d, m' is divisible by m. If d and d' have no common prune factor other than 2 or 5, the number of digits in the period for 1/dd' is the 1. c. m. of m, w'. Hence it suffices to know the length of the period for l/p", where p is a prime. If l/p has a period of m digits and if l/pn is the last one of the series l/p, l/p2, . . . which has a period of m digits, then the period for l/pa for a>n has mpa~~n digits. For p = 3, we have n=2; hence l/3r for r^2 has a period of 3r~2 digits. For any prime p for which 7^p^l01, we have n = l, so that l/pa has a period of mp""1 digits. Note that l/p and l/p2 have periods of the same length to base 6 if and only if ft"""1 = 1 (mod p2) . Proof is given of Catalan's30 first theorem, which holds only when 10m^l (mod JD), i. e., when m is not a multiple of the number of digits in the period. For example, the sum of the Mh and (6-|-fc)th remainders for 1/13 is not a multiple of 13.
E. Prouhet32 proved Thibault's31 theorem on the period for l/pn. He32' noted that multiples of 142857 have the same digits permuted.
P. Lafitte33 proveci Midy's27 theorem that, if p is a prime not dividing m and if the period for m/p has an even number of digits, the sum of the two halves of the period is 9 ... 9.
J. Sornin34 investigated the number m of digits in the period for 1/D, whereD is prime to 10. Theperiodisz=(10m-l)/D. First, let D = Then x = IQy — 1 , where
- - --
Finally, we reach v— |1 — ( — k)m\/D, and x is an integer if and only if v is. Hence if we form the powers of the number k of tens in Z>, add 1 to the odd powers, but subtract 1 from the even powers of k, the first exponent giving a result divisible by D is the number m of digits in the period.
s°Nouv. Ann. Math., 1, 1842, 464-5, 467-9.
»76wf., 2, 1843, 80-89.
"Ibid., 5, 1846, 661.
""Ibid., 3, 1844, 376; 1851, 147-152.
™ThiH .. 3Q7-Q f!f Am*»r MatJh A/T^nf/hlv 1O 1Q10I im_O