316 



REPORT 1875. 



c. The like table for the root 10 for all prime numbers between 5000 and 

 15000 (no column for w, nor any prime root given), pp. 53-61. 

 A specimen is 



p p—1 e n 



9859 2 . 3 . 31 . 53 3286 3 



viz. mod. 9859 we have lO^-""^!. Eut in a large number of cases we have 

 H=l, and therefore 10 a prime root. Tor example, 



9887 2.4983 9886 1. , 



23. For a composite number n, if N='nr(n) be the number of integers less 

 than 11 and prime to it, than if x be any number less than n and prime to 

 it, we have x^=l (mod. n). Eut we have in this case no analogue of a 

 prime root — there is no number x, such that its several powers x^, x'-, .... 

 x^~'^ (mod. n) are all different from unity ; or, what is the same thing, there 

 is for each value of x some submultiple of N, say N', such that ■-i;^''=1 

 (mod. n). And these several numbers W have a least common multiple I, 

 which is not =]Sr, but is a submultiple of N ; and this being so, then for all 

 the several values of x, I is said to be the maximum indicator. For instance, 

 91 = 12, N='nT(ii); the numbers less than 12 and prime to it arc 1, 5, 7, 11. 

 We have (mod. 12) l'=l, 5=^1, 7'=1, ir = l, or the values of W arc 1, 

 2, 2, 2; their least common multiple is 2, and we have accordingly 1=2: 

 viz. x-=l (mod. 12) has the «7(12) roots 1, 5, 7, 11. So n=24, ■vj{n)=S; 

 the maximum indicator I is in this case also =2. 



A table of the maximum indicator Ji^l to 1000 is given 



Cauchy, Excr. d'Analyse &c. t. ii. (1841), pp. 36-40, contained in the 



" Momoirc sur la resolution des equations indeterminees du premier degre en 



uombrcs entiers," pp. 1-40. 



24. It thus appears that for a composite number n, the ^(m) numbers less 

 than n and prime to it cannot be expressed as ^ (mod. n) to the power of a 

 single root ; but for the expression of them it is necessary to employ two 

 or more roots. A small table, n=l to 50, is given 



Cayley, Specimen Table M^o"6^ (mod. N) for any prime or composite 

 modulus ; Quart. Math. Journ. t. ix. (1867), pp. 95, 96, and folding sheet. 

 A specimen is 



