240 report — 1859. 



and, whichever oF them we take, there will be — ^^r primitive roots, fof 



which b will have the index 1(3', while a retains the index a'. We must next 

 form the residues of the X— a products included in the formula a* by; where 

 x has any value from 1 to a inclusive, and y any value from 1 to B— 1. These 

 residues are all incongruous ; the indices of all of them are known ; and, 

 together with the a powers of a already entered in the table, they exhaust all 



»— 1 

 the numbers which have indices divisible by£— — -. 



X 



In practice, it will almost always happen that X is equal to p— 1. When! 

 this is so, nothing remains to complete the operation but to enter in the 

 Table the residues of the numbers a* by opposite to the indices corresponding 



to them. But, if X </>—!, we may take that residue which has — r— for its 



index, and use it to replace a in the preceding operation, while b is replaced 

 by some other residue not yet entered in the Table. In this way we shall 

 ultimately (and in practice very speedily) obtain a complete Table of Resi- 

 dues corresponding to given indices, which, of course, immediately supplies 

 us with the inverse Table of Indices corresponding to given residues. It 

 will be seen (as has been already observed) that the process is not dissimilar 

 to Gauss's method for determining a number appertaining to the exponent X 

 when we already know two numbers a and b appertaining to the exponents; 

 a and /3 respectively. But it is so arranged by Jacobi that hardly a single 

 figure is wasted, the primitive root, instead of being found by a preliminary 

 investigation, presenting itself at the end of the operation, and being recog- 

 nized by its standing opposite to the index 1. 



To calculate with rapidity the residues of the powers of a number, Jacobi 

 employs a method proposed by M. Crelle in his Journal, vol. ix. p. 30, and 

 which is most easily explained by an example. 



Let jt>=ll, and let it be required to determine the residues of the powers 

 of 3 ; and the residues of those powers multiplied by 7. 



Column I. 1,2,3,4,5,6, 7,8,9,10 

 „ II. 3,6,9, 1,4,7, 10,2,5, 8 



III. 3, 9, 5, 4, 1, 



IV. 10, 8, 2, 6, 7. 



The first column contains the numbers 1, 2, 3 . . p—1. The second- 

 column begins with 3 (the number the powers of which we are considering)^ 

 and consists of numbers formed by successive additions of 3, multiples of 1 1 

 being rejected as fast as they arise. The third column also commences 

 with 3, and is so formed that any number r in it is followed by the number 

 which in column II. stands under r in column I. This column contains the 

 residues of the powers of 3 taken in order, and stops at 3 s because after that 

 the same residues recur. Lastly, column IV. begins with 10 (the number 

 which in column IT. stands under 7 in column I.), and is formed in the same: 

 way as column III. It represents the residues of 7.3, 7.3 2 , &c. ... 



15. Quadratic Residues. — It appears from the theorems cited in Art. 12,- 

 that the numbers 1, 2, 3,... p—1, divide themselves into two classes of Qua- 

 dratic Residues, and Quadratic non-Residues, comprising -| (p — 1) numbers- 



P-i 

 each. Every quadratic residue a satisfies the congruence a; 2 =l,modp;: 



P-i 

 every quadratic non- residue b satisfies, instead, the congruence x 2 = — ], 



