200 Mr. W. Shanks on the Period of the [Feb. 19, 



III. " On the Number of Figures in the Period of the Reciprocal 

 of every Prime Number below 20,000." By WILLIAM 

 SHANKS. Communicated by the Rev. GEORGE SALMON. 

 Received December 2, 1873*. 



The following Table, in reality the joint production of the Rev. George 

 Salmon, F.R.S., and myself, was commenced, and indeed nearly com- 

 pleted, before either calculator was distinctly aware that Burckhardt, 

 Jacobi, or Desmarest had written or published any thing on the same 

 subject. This fact is perhaps to be regretted ; but it has led to the in- 

 dependent recalculation, by two different methods, both of Burckhardt's 

 ( Jacobi's Table is professedly a reprint of Burckhardt's) and of Desmarest's 

 Table, and has resulted in the detection of several errors, which have, as 

 far as I know, never before been pointed out. These errors, in the first 

 place regarded as discrepancies, have been carefully examined ; in fact 

 every case has been reworked by me, with the view of either proving or 

 disproving the accuracy of such numbers as differ from those in our 

 Table. The result is, that such discrepancies are found to be errors both 

 in Burckhardt and Desmarest. The two lists of errors are given below. 

 I now proceed to give the theorems used, and some account of the 

 means employed by me in forming the Table. 



Let P be any prime number, except 2 and 5. Then, from Fermat's 



KF- 1 

 theorem, we have p =1 ; or, adopting the usual notation, 10 P ~ 1 ^1. 



Again, since the number of figures in the period of the reciprocal of 

 all primes is not P 1 (or, in other words, since 10 is not a primitive 



root of all primes), 



p-i 

 Let 10 =1, where n is even or odd, not less than 2, and not greater 



P-l 



than o * Then we have 



Zi 



(1) The number of figures in the period of the reciprocal of P is either 

 P 1 or a submultiple of P 1. -, Q B 



(2) Let a and b be integers, and let m be the remainder from -p > 

 that is,. let 10"=m; then 10 ab =m*. 



In practice b is never greater than 2, at least little or no advantage is 



P 1 

 gained by putting b higher. Also ab need not be greater than ^ ' 



Cor. "When m is greater than -Q we may obviously use P m, or 

 simply m; for (P m) 2 =P 2 2Pm+m 2 =m 2 , or, because ( m) 2 =m 2 , 

 b being 2. 



(3) Let 10 a =m, and I0 b ~n ; then 10 a +*=mn. 

 In practice a+b is never greater than P 1. 



Cor. 1. When m and n are each of them less than P, we may with 

 advantage use m and n; that is, we may subtract m and n severally 

 from P ; for (P-m)(P w)=P 2 -P(w + w)+mn=( m)( n). 

 p ^ P 1 



Cor. 2. When m is > 5- , and n is < Q , or vice versa, we may use 



* The part from 17,000 to 20,000 was received January 8, 1874. 



