ON MATHEMATICAL TABLES. 835 



II. p = 7m — l. (1) gives for the several prime numbers of this form 13, 

 41 ... . 937 the congruence roots, mod. jj ; for instance, 



and (2) gives under the simple and the primary forms the prime factors /(rj) 

 of these same numbers 13, 41 ... . 937 ; for instance, 



p f(ri) simple. f(r]) primary. 



13 Vo + ^r,, ' 3 + 7„, 



41 4+ ,,„ -U + 7r,,-7tj,. 



Thus lS=(T]^ + 2T].Xvi + 2riJ(ri, + 2ri^), as is easily verified; the product of 

 first and second factors is =4:+3r]^ + Sri^ + 5r)^, and then multiplying by the 

 third factor result is 42 + 29(ri^ + ri^, =13. 



III. p = 7m-\-2 or 7m + 4. (1) gives for the several prime numbers of this 

 form 2, 11, .... 991 the congruence roots, mod. p ; for instance, 



P Vo ni 

 2 0-1 

 11 4 -5 



and (2) gives the primary prime factors /(jj) of these same numbers ; — 



F f(v) 



2 Vo 



11 1-2^1. 



IV. ^3 = 7m + 3 or 7m +5. The prime numbers of these forms, viz. 3, 5, 

 17, 19 ... . 997, are primes in the complex theory, and are therefore simply 

 enumerated. 



The arrangement is the same for the higher prime numbers \ = 23 &c., 

 for which ideal factors make their appearance, but it presents itself under a 

 more complicated form. Thus \ = 23, \ — 1 = 2 . 11, and the factors of 22 are 

 22, 11, 2, 1, There are thus four sections. 



I. a a prime root, or a"^ + a"'- • • • +a- + a-\-l = 0. 

 II. r)^=a + a~\ .... j;jg = a" + a~", or tj a root of 

 7," + »,"-10,,' .... +15,,= -6^-l. 

 III. T}^ = a + a.-, jj,= a~^ + a~^ or ?; a root of rj^+ri+Q = 0. 

 rV. Eeal numbers. 



I. p = 23m+l gives for the prime numbers of this form 47, 139. . . . 967 

 congruence roots mod. ^3 and also congruence roots mod. p^*; these last 

 in the form a + bp + cp^, where a is given in the former table; thus first 

 table:— ^ „ „2 „3 _ ^22 



47 6 -11 -19 +8; 



and second table — 



p a a* a' . . . .a^ 



47 +p-2p- +13p-23/ +19p-8/ +22w + 22/, 



The meaning is that, p = 47, the roots of the congruence 



ar + a"'+ .... +a- + a + l40 (mod. 47') 

 are a= G +p-2p\ a'= - 11 + 13jj - 23/, &c. 



* Where, as presently appearing, 3 is the index of ideality or power to which the ideal 

 factors have to be raised in order to become actual. 



