1874.] Reciprocal of every Prime Number below 20,000. 201 



w and n, or vice versa, obtaining a negative result, which becomes 

 positive by being subtracted from P. p ., 



(4) Let 2c and 3c, not greater than ^ , be submultiples of P 1 ; 



and let 10<~ + S, and 10 2c =S - 1 ; then 10*== + 1 . This is evident from 

 (2) and (3). ^ 



From (1) we have 10 2 =1, according as the submultiple of P-l is 

 even or odd. 



On these theorems and adjuncts my calculations have been based. 



10 P-1 



They enable us to find the remainder either from p . or from anv 

 p-i 



submultiple, such as p , or from any figure in -~- , and, if required, 



the figure itself. Compared with other methods, such for instance as 

 Dr. Salmon's *, mine may seem tedious, requiring as it does much multi- 

 plication and division. All I can say is, I did not find it so, though I 

 am free to admit that the calculation of such a Table as ours demands 

 very considerable labour. 



It would be foreign to my purpose to enter upon the consideration of 

 primitive roots, or even of prime numbers. If we have found 10 to be a 

 primitive root of a great many prime numbers between 10,000 and 

 20,000, we have contributed something, as far as I know, quite new. 

 In addition to this we have found the number of figures in the period of 

 each of the other primes between 10,000 and 20,000, and have corrected 

 upwards of 70 errors in Burckhardt's and Desmarest's Tables. 



I beg to refer to the works of Euler, Lagrange, Legendre, Gauss, Poinsot, 

 Cauchy, and Jacobi (mentioned by Desmarest), and to Desmarest himself, 

 for valuable information touching prime numbers and primitive roots. 



I cannot, however, refrain from quoting from Desmarest's ' Theorie des 

 Nombres ' the view of Euler as to prime numbers and primitive roots : 

 " On ne peut saisir entre un nombre premier et les racines primitives qui 

 lui appartiennent, aucune relation d'ou Ton puisse deduire une seule de 

 ces racines, de sorte que la loi qui regne entre elles parait aussi profonde- 

 ment cachee que celle qui existe entre les nombres premiers eux-memes." 



Not discouraged by Euler's remark, Desmarest thus writes : " Car 

 pourquoi nous serait-il defendu d'ajouter que nous croyons que 1'intel- 

 ligence humaine n'a pas, sur ce point, dit son dernier mot, et que les 

 operations nombreuses que nous avons du faire sur les nombres, ne nous 

 ont pas convaincu de 1'impossibilite de saisir, sinon 1'ensemble, du moins 

 quelques-uns des anneaux de la chaine mysterieuse qui unit les racines 

 primitives aux nombres premiers." 



* Note by Dr. Salmon. The method here referred to is explained, ' Messenger of 

 Mathematics' (1872), p. 49. It is founded on the remark that if we have 10 a ^2^, 

 10*=2, we may deduce 10"*- 6 ^ = 1. Thus, let the prime be 251, we can at once write 

 down the equations lCfe-2 2 , 2=10 T , whence immediately 10 23 = -1, 10 50 =1. 

 In like manner from the equations 10=2'3, 10 ft =2'3?, 10 C ==2 M 3 P , we deduce that 

 the number of figures in the period of the reciprocal of the prime is 



a(mrnq) +b(nprl) +c(lq_ mp). 



By the application of these principles I calculated the results obtained in the fol- 

 lowing Table as far as 18500. For the primes above that number Mr. Shanks is solely 

 responsible; but my experience of his accuracy gives me confidence in his result?. 



VOTy. XXII. 



