ON MATHEMATICAL TABLES. 313 



The object of the table is explained in the ' Bemerkungen,' p. 499, by 

 Hchoring, the editor of the volume, viz. it is to facilitate the calculation of 

 the circular arcs the cotangents of which are rational numbers. To take a 

 simple example, it appears to be by means of it that Gauss obtained, among 

 other formulie, the following : 



and 



^ =12 arctan :r^ + 8 arctan -^—5 arctan ---, 

 4 lo 57 isoy 



= 12 arctan -^ +20 arctan ^ + 7 arctan 239+24 arctan _. 



Art. 2. [F. 12. Divisors ^-c] continued. Prime Boots. The Canon Arithmeticus, 



Quadratic residues. 



17. Prime Boots. — Let ^j be a prime number; then there exist ra-f^) — 1) 

 inferior integers g, such that all the numbers 1, 2, . . .p— 1 are, to the mo- 

 dulusp, ^ l,g,r/", . ..gP~^ (9^~^ is of course ss 1); and this being so, cj is 

 said to be a prime root of p ; and moreover the several numbers ^«, where a. 

 is any number whatever less than and prime to ^3 — 1, constitute the series of 

 the m(p — 1) prime roots of j). It may be added that if /3 be an integer num- 

 ber less than j; — 1, and having with it a greatest common measure =Jc, so 



p-l jS 



that {gP) ^ ^ g^ , s^ 1, 1 since ^ is an integer, and gP~^:^l\ then g^ has 



p— 1 . . . 



the indicatrix , : the prime roots are those numbers which have the indi- 



catrixp — 1. 



The like theory exists as to any number N of the form p'" of 2p'". 



There ai-e here 'm(E), =]Sr(l — -j or ^Nll— -j (in the two cases respec- 

 tively) numbers less than N and prime to it ; and we have then 'm('w(J^y\ 

 numbers <7 such that to the modulus N all these numbers are :=1, (^, ^-. .. 

 (ynr(N)-i ((^to(N) jg gf courso H3^ 1) ; and this being so, g may be regarded as 

 a prime root of N (=^'" or 2jj'» as the case may be) ; and moreover the several 

 numbers <y*, where a. is any number whatever less than and prime to 'nr(N), 

 constitute the series of the ct ('«b-(jS')) prime roots of N. Thus 1^=3-= 9, 



w(N)=6; we have 



1, 2', 2\ 2\ 2\ 2' 



= 1, 2, 4, 8, 7, 5 mod. 9 ; 

 or prime roots are 2^ and 2', =2 and 5. 

 So also N=2. 3^=18, ^N) = 6; we have 



= 1, 5, 7, 17, 13, 11 mod. 18; 

 and 5' and 5% =5 and 11 arc the prime roots of 18. 



18. A small table of prime roots, p= 3 to 37, is given 



Euler, Op. Arith. Coll. t. i. pp. 525-526. The Memoir is entitled 

 " Demonstrationes circa residua e divisione potestatum per numeros primes 

 resultantia," pp. 516-537 (1772). 



19. A table, 2^ andjj'", 3 to 97, is given 



Gauss, ' Disquisitiones Arithmeticse/ 1801 ("Werke, t. i. p. 468). This 



