1904.] Corrigenda in Tables of Reciprocals of Prime. 359 



" Corrigenda in Mr. W. Shanks's Tables ' On the Number of 

 Figures in the Eeciprocal of a Prime.' " By Lieut.-Oolonel 

 Allan Cunningham, RE., Fellow of King's College, London. 

 Communicated by the Secretaries. 



This paper contains the result of an examination of the late Mr. W. 

 Shanks's MS. Tables, with above title, now deposited* with the Eoyal 

 Society. These tables have been collated^ with the following printed 

 tables, — 



1. ' Periodische Dezhnalbruche,' by H. Bork, Berlin, 1895. The Appendix 

 (pp. 36 — 41) contains a Table (computed by Dr. F. Kessler) giving the Residue- 

 Index (q), i.e., tbe maximum divisor yielding 10 (p— l ) + 1 E + l (mod. p) for every' 

 prime (p) ^> 100,000, for whivh q > 2. 



2. " Periode des Dezimalbruclies fur Ijp wo p eine Primzahl," by H. Hertzer,. 

 printed in Grunert's ' Archiv der Math, und Phys.,' vol. 2, 1902, p. 249. The 

 Table (pp. 249 — 251) is a continuation of the preceding Table for primes up to' 

 p ^> 112,400, and is arranged in the same manner. 



Shanks's MS. Tables give the period-length (say £) of Ijp for all 

 primes from 30,000 to 120,000. The collation was effected by simply 

 multiplying Shanks's value of f by Kessler's or Hertzer's value of q - y 

 the product of £q should = (p - 1) in every case. The collation was, 

 of course, only possible for such primes as have q > 2 (being the only 

 ones shown by Kessler and Hertzer), thus excluding about two-thirds 

 of the total number of primes ; the £, q of these excluded primes are r 

 however, easily computed when required. 



By this collation a number of discrepancies (102 in all) were 

 discovered between Shanks's MS. and the printed German tables. 

 The values of £ have in all these cases been re-computed,] with the 

 result of detecting errata, as follows : — 



Shanks p missing 5 • £ wrong 66 ; total 71 "i 



Kessler p missing 8; q wrong 20; total 28 > Total 102. 



Hertzer p missing 1 ; q wrong 2 ; total 3 J 



* Part of the MS., viz., for primes from 30,000 to 60,000, is bound up with a 

 small volume marked Constants and Primes (with the Press-mark 103 d 15), and 

 the rest, viz., for primes from 60,000 to 120,000, is bound up with the ' Eoyal 

 Society Archives,' vols. 60, 61. Mr. Shanks's MSS. bear dates as follows : — 

 For primes 30,000 to 60,000, dated 1875. 

 For primes 60,000 to 75,000, dated 1876. 

 For primes 75,000 to 110,000, dated 1877. 

 For primes 110,000 to 120,000, dated 1880. 

 f By the writer of this paper, with the help of an assistant (Miss E. Cooper) 7 . 

 by permission of the Council of the Eoyal Society. 



% By the writer himself, and verified in part by Miss E. Cooper. 



2 c 2 



