382 Mr. W. Shanks on the [June 18, 



wP 2 , and so on. Since the period of = 18, and since the remainders 



19 



resulting from dividing 18 such periods successively by 19 are, in order, 15, 

 11, 7, 3, 18, 14, 10, 6, 2, 17, 13, 9, 5, 1, 16, 12, 8, 4, 0, it follows that 



- = 18x19=342. The law of such remainders, after the first has 

 19 



been obtained, is simple enough, and may be written down at once. 



Again, since the period of = 81, also since the remainders resulting 

 loo 



from dividing 163 such periods, each of 81 figures, successively by 163 



are, in order, 149, 135, 121, 107, 93, 79, 65, 51, 37, 23, 9, 158, 144 



(the series consisting of 163 terms, of which the last is 0), it follows that 



1 1 = Six 163 = 13203. The law of the above series is evident, and 



163 2 



the number of terms is easily found to be 163. There is an 

 obvious exception when P = 3; then the period is divisible by P, 

 and the number of figures in the reciprocal of 3 2 is 1, of 3 3 is 3, and of 

 3 W is 3 n ~ 2 . There are other exceptions also, or at all events one. 

 Desmarest, for instance, has remarked that in the case of P=487, the 

 period is divisible by 487 ; and therefore the number of figures in the 

 reciprocal of 48 7 2 is the same as that in the reciprocal of 487, viz. 486. 

 I am not acquainted with the general theory of such exceptions ; nor do 

 I know what other primes (if any) besides 3 and 487 have the same 

 peculiarity. 



"With these explanations the following Table can readily be under- 

 stood. We mark with an asterisk those cases in which the resolution is 

 complete, thus 28 | 29 . 281 . 12149 9449. "We are to be understood 

 as affirming that 12149 9449 is a prime number. 



Primes, Prime Factors, &c. 



