[ 369 J 

 LI. Remark on Primitive Radices. By the Rev. R. Murphy*. 



TJTAVING had the honour of receiving from M. Jacobi his 

 ** * Canon Arithmeticus, sive tabulae, quibus exhibentur 

 pro singulis numeris primis, vel primorum potestatibus infra 

 1000, numei i ad datos indices et indices ad datos numeros per- 

 tinentes, Berolini, 1839,' I sought in the tables for some law 

 relative to primitive radices. Being desirous of completing a 

 work, entitled ' New Forms for the calculation of Logarithms,' 

 which is nearly ready for the press, I could not bestow suf- 

 ficient time to investigate the absolute generality of the fol- 

 lowing properties; yet even as empiric forms easily recollected, 

 and verified in every case in those tables, and also in every 

 case where I have taken small primitive radices, they will be 

 found useful ; at the same time the attention of mathematicians 

 will, I hope, be drawn to the demonstration of the generality of 

 these forms (or the contrary), though belonging to what is usu- 

 ally considered a dry subject, " The Theory of Numbers." 



In page 6, we find a table, " Numeri primi, quorum 10 

 radix primitiva est," viz. 7, 17, 19, 23, 29, 47, 59, &c. to 

 2543 ; and the table from which this is deduced is (with some 

 exceptions) carried as far as 9901. The following prime 

 numbers, to which 10 is primitive radix, may be culled from 

 those in Burckhardt's table, viz. 7, 17, 47, 97, 1G7, 257, 367 

 (497 is not prime), 64? 7 (817 is not prime), (1007 is not 

 prime), 1217, 1447, 1697 (1967 is not prime). Hence every 

 prime number of the form 10 w 2 + 7 has 10 for a primitive 

 radix, as far as can be verified by these tables. 



As this property, if general, must depend on some relation 

 between the numbers 7 and 10, 1 remarked that 7 was the only 

 prime number less than 10, to which the latter was the pri- 

 mitive radix; accordingly, I tried in several instances the 

 more general formula an 2 + p, where p is a prime number less 

 than a number, a to which a is a primitive radix (as 3 n 2 + 2, 

 5n 2 + 3, &c.) ; from the results it would appear to be a 

 general property, that under those circumstances a w 2 + p 



I when prime, and p greater than - ) will have a as a pri- 

 mitive radix. From the cause I have mentioned, I have not 

 bestowed that attention to these properties I could have 

 wished ; the attention of others will now be called to them ; 

 in the mean time they may stand as empiric formulae. 



London, October 1841. Robert Murphy. 



* Communicated by the Author. 



Phil, Mag, S. 3. Vol. 19. No. 125. Nov, 1841. 2 B 



