1878.] 



llr Glaisher, On circulating decimals. 



201 



487^ 



I have also verified that the latter period is not divisible by 

 487, so that the period of (4^)^ contains 486 x 487 digits. 



Since ^ lO^'"^ - 1 = (10^*^ + 1) (10'^' - 1) 



and lO'^'' + 1 = (10^^ + 1) (-10"' - 10=^' + 1) 



it follows that 10^'^'- 10'^ + 1 is divisible by 487 and by (487)'. 

 The quotients thus obtained are given by Mr Shanks in vol. xxv., 

 p. 553, of the Proceedings of the Royal Society (1877). The prime 

 69499 has a period of 486 digits, and therefore ^^ {W - 10^' + 1) 

 and (^i^)' (10'*^' - 10'' + 1) are both divisible by 69499 : this latter 

 quotient is also given by Mr Shanks. 



The fact that 487 is such that 10'''= 1 (mod 487') is interest- 

 ing for the following reason. In t. III. (1828), p. 212, of Crelles 

 Journal, Abel proposed the question, " Kanii o.'^'^ — 1, wenn jx eine 

 Primzahl und a eine ganze Zahl und kleiner als [x und grosser 

 als 1 ist, durch fj? theilbar sein?" On pp. 301 — 302, Jacobi 

 answered this question by showing that 3'° = 1 (mod 11^), so that 

 9'''-l (mod ll'O, and also that 14^^=1 (mod 29") and 18^^=1 

 (mod 37^). The case of 487 is a solution of Abel's congruence 

 a'^"' = 1 (mod /A^), /i > a, when a = 10, and is the only known solu- 

 tion. There seems no reason to suppose that there are not other 

 solutions, and that the congruences 10^"-^= 1 (mod |U-^), &c., may not 

 have solutions. The next solution above 487 of the congruence 

 lG'^"'s 1. (mod jx^) may be a very high number, as is evident by 

 merely considering the diminution of the chance of a number 

 dividing exactly its own period — the latter being regarded merely 

 as a number taken at random — as the number increases and 

 consequently the number of possible remainders increases ; the 

 j)eriod being of course divisible by the number only in the case 

 when the last remainder is zero. A computer finding the rule 

 in § 2 (x) to be true for all the numbers except 3, to which 

 he applied it, might believe it to be universally true, with this 

 sole exception, and actually use it in forming a table such as 

 that at the end of this paper. This affords a good instance of 

 the necessity, in the case of a table in Theory of Numbers, of 

 fully explaining the mode of construction ; as nothing can be more 

 unsafe than the use of empirical rules in the Theory of Numbers 



15—2 



