114 Interesting Properties of Numbers. 



these quotients mid remainders will satisfy the conditions of equa- 

 tions (9) and (12). 



We know by the Theory of Numbers, that the remainder of 



P-1 



N 2 divided by P is either 1 or P — 1, Hence, it follows that 



when the remainder is P — 1, the number of terms in the periods 



will beV — 1 or a submultiple o/ P - 1. And ivhen the remain- 



P — 1 



der is 1, the nimiber of terms in the periods m,ust be or else 



ii 



P — 1 



a submultiple of 



If N is a composite number of the form x"-, ^'\ y", &c, when x, 

 ^, 7, &c. are prime factors, and a, b, c, &c. are whole numbers, 

 and P is also a composite number, whose prime factors do not 

 differ from those which compose N, then the process will termin- 

 ate ; for X can be so taken as to make N^ divisible by P without 

 a remainder. 



If P, besides containing the prime factors common to N, con- 

 tains other prime factors, the process will not terminate, but must 

 give periods of quotients and remainders ; but in this case, other 

 terms will occur before the periods commence. 



If N and P are both primes, the one of the form An-\-l, and 

 the other of the form An-\-3, we know by the law of reciprocity 

 of primes, that if the remainder ?'p ^ is P — 1, then also will the 



~2~ 

 remainder rp _ , be P— 1, when N and P exchange places; so 



that the number of terms in the periods in the first case, will be 



P - 1 ; and in the second case, N — 1. 



We will now illustrate these singular properties by numerical 



results. If N~20 and P = 37, we shall have as follows : 



auotients^ 0,10,16, 4, 6, 9,14,11,17,16,15, 2,14, 1, 1,12, 8,12 

 ^ il9, 9, 3,15,13,10, 5, 8, 2, 3, 4,17, 5,18,18, 7,11, 7 



T?pmn;n,1pv« 5 20, 30, 8, 12, 18, 27, 22, 33, 31, 28, 5, 26, 2, 3, 23, 16, 24, 36 

 itemainaeis ^ ^^^ ^^ ^9, 25, 19, 10, 15, 4, 6, 9, 32, 11, 35, 34, 14, 21, 13, 1 



We have arranged the quotients in two horizontal lines, so that 

 the ^^ is directly over the ^-p _ w quotient; in this arrange- 



4-x 



2 



ment, we more readily see that they satisfy the condition (12); 

 the remainders we have arranged in a similar manner. 



