ON MATHEMATICAL TABLES. 315 



21. There is given, 



Jacobi, Cjcellc, t. xxs. pp. 181, 182, a table of m! for the argument m, 

 such that 



l-\-g'"=rf'' {moA. p.), ^=7 to 103, and to = to 102. 



A specimen is 



. to 103 



for instances = 19, l-|-10''=10i^ (mod. 19). 



Jacobi remarks that this table was calculated for him by his class during 

 the Avinter course of 1836-37 ; and that, by means of the since-published 

 ' Canon Arithmeticus,' the same might easily bo extended to all primes under 

 1000. In fact for any such number p, putting any number of the table 

 " Indices "=m, the next following number of the table gives the value of m'. 



22. We have next in Reuschle's Memoir {ante, No. 15) the folloudng 

 relating to prime roots 



C. Tafeln fiir primitive Wurzeln imd Hauptexponenten, oder V. erweiterto 

 und bereicherte Burkhardtsche Tafel, pp. 41-61, being divided into tkree 

 parts ; viz. these are 



a. Table of the Hauptexponenten of the six roots 10, 5, 2, 6, 3, 7 for all 

 prime numbers of the first 1000, together with the least primitive root of 

 each of these numbers (pp, 42-46). 



A specimen is as follows : — 



10 5 2 6 3 7 w 



P P 



— 1 e n en en en en en 



53 2-. 13 13 4 52 1 52 1 26 2 52 1 26 2 2 



where e is the Haupt-exponent or indicatrix of the root (10, 5, 2, 6, 3, 7, as 



p-1 

 the case may be), n=- , w the least piimitive root ; thus 



^=53, 10^^=1, 5°^=1, 2^^^1 



(2 being accordingly the least prime root), 



6''=1, 3"=1, 7^'=1. 



The number w of the last column is the least primitive root ; it is, of course, 

 not always (as in the present case) one of the numbers 10, 5, 2, 6, 3, 7 to 

 which the table relates: the first exception is p=191, 111=19, the highest 

 value of ty being it; = 21 corresponding to j) =409. 



b. The like table for the roots 10 and 2 for all prime numbers from 1000 

 to 5000, together with as convenient as possible a prime root (and in some 

 cases two prime roots) for each such number (pp. 47-53). 



A specimen is : — 



10 2 



p p — 1 en e n tu 



1289 2^7.23 92 14 1618 6,11 



viz. here mod. 1289, \Q'-=1, 2'"^1 ; and two prime roots are 6, 11. We 

 have thus by the present tables a prime root for every prime number not 

 exceeding 5000. 



