ON MATHEMATICAL TABLES. 



317 



viz. for the modulus 12 the roots are 5, 7, having respectively the indicators 

 2, 2, viz. o'^^l (mod. 12), 7"=^1 (mod, 12). Hence also the maximum indi- 

 cator is =2. 0( = 'E7(n))=4 is the number of integers less than 12 and prime 

 to it, viz. these are 1, 5, 7, 11, which in terms of the roots 5, 7 and to mod. 12 

 arc respectively =s o^T, b\T, b°.l\ and 5\7'. 



25. Quadratic Residues. — In regard to a given prime number jj, a number 

 N is or is not a quadratic residue according as the index of N is even or odd, 

 viz, g being a prime root and N^^/", then according as a is even or odd. 

 But the quadratic residues can, of course, bo obtained directly without the 

 consideration of prime roots. 



A small table, ^=3 to 97 and N= — 1 and (prime values) 3 to 97, is given 



Gauss, DisquisitionesArithmeticoe, 1801; Table II. (Werke, t. i. p. 469): 

 I notice here a misprint in the top line; it shoiild be —1, +2, +3, &c., 

 instead of 1, +2, +3, &c. ; the —1 is printed correctly in p. 499 of the 

 French translation ' Recherches Arithmetiques,' Paris, 1807. 



Ji. specimen is 



19 



&c. 



viz. — 1, 2, 3, 13 are not, 5, 7, 11, 17 &c. are residues of 19. The residues 

 taken positively and less than 19 are, in fact, 1, 4, 5, 6, 7, 11, 16, 17. 



The same table carried fromp = 3 to 503, and prime values N=3 to 997, 

 is given 



Gauss, Werke, t. ii. pp. 400-409. Specimen is 



19 



2 



3 



11 



13 



17 



19 23 



&C. 



viz. the arrangement is the same, except only that the — 1 column is omitted. 



26. We have also by Gauss 



Table III. Disquisitioues Arithmeticae (Werke, t. i. p. 470), for the con- 

 version into decimals of a vulgar fraction, denominator p or p*^, not exceeding 

 100. The explanation is given in art, 314 et seq. of the same work. 



But this table, carried to a greater extent, is given by Gauss, Werke, 

 t. ii. pp. 412-434, " Tafel zur Yerwandlung gcmeiner Briiche mit Nenuern aus 

 dem erstcn Tausend iu Decimalbriiche ;" viz. the denominators are hero 

 Xirimes or powers of primes, p*^ up to 997. 



To explain the table, consider a modulus p*^ (where /j. may bo = 1) ; 

 if 10 is not a prime root of p'^, consider a prime root r, which is such that 

 r'lO^ (mod. j**^), e being a submultix^le of ^^'^"'(p — 1); say we have 



ef=li''-\p-l), then 10^=1 (mod. p''). 



N" 



Consider any fraction — , then 



we may write N^?-^'+' (mod. p'^) h from to /—I, and I from to e—1, 



N lOV 



^ \0V, and consequently —^ and — ^ have the same mantissa (decimal part 



regarded as an integer) ; hence, in order to know the mantissa of every frac- 



N" rl 



tion whatever of -77, it is sufficient to know the mantissa of —^, 

 p'^ p'^ 



that is the 



