314 



KEPOKT 1875, 



gives ia each case a prime root, and it shows the exponents in regard thereto 

 of the several prime numbers less thaup or p'". Thus a specimen is 



2 . 3 . 5 . 7 . 11 . 13 . 17 . 19 . 23 . 29, &c. 



viz. for 27 we have 2 a prime root, and 2=2', 5=2% 7 = 2^ 11 = 2'\ 

 &c. ; and so also for 29 we have 10 a prime root and 2^ 10", 3ss 10", 

 5 = 10^&c. 



20. Small tables are probably to be found in many other places ; but the 

 most extensive and convenient table is Jacobi's ' Canon Arithmeticus,' the 

 complete title, of which is 



' Canon Arithmeticus sive tabula quibus exhibentur pro singulis numeris 

 primis vel primorum potestatibus infra 1000 numeri ad dates indices ct 

 indices ad dates numeros pertinentes.' Edidit C. G. J. Jacobi. Berolini, 

 1839. 4°. 



The contents are as follows : — Page 



Introductio i to xl 



Tabulse numerorum ad indices dates pertinentium et indicum 

 numero dato correspondentium pro modulis primis minoribus 



quam 1000 ._ 1-221 



Tabulae residuorum et indicum sibi mutuo respondentium pro 

 modulis minoribus quam 1000 qui sunt numerorum pri- 

 morum potestates 222-238 



Hujus tabula ea pars quae pertinetad modules formae 2", inve- 



nitur 239-240 



The following is a specimen of the principal tables : — 



P = 19, p-l = 2.3\ 

 Numeri. 



where the first table gives the values of the powers of the prime root 10 (that 

 10 is the root appears by its index being given as =1) to the modulus 19, 

 viz. 10' ^ 10, 10- EEE 5, 10^ ^ 12, &c. ; and the second table gives the index 

 of the power to which the same prime root must be raised in order that it 

 may be to the modulus 19 congruent with a given number, thus 10'*=sl, 

 10'^ ss 2, 10' ^ 3, &c. The units of the index or number, as the case may 

 be, are contained in the top line of the table, and the tens or hundreds and 

 tens in the left-hand column. 



