498 
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 
[611 
IV. p = 7m + 3 ov 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 A = 23, &c., for which 
ideal factors make their appearance; but it presents itself under a more complicated 
form. Thus A = 23, A —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 22 4- a 21 4- . • • + a 2 4- a 4- 1 = 0 ; 
II. 7j 0 = a 4- a -1 ,..., r) 10 = a 11 + a -11 , or rj a root of -?? 11 4- rj w — 10?7 9 +...4 lorf — (jtj — 1 = 0 ; 
III. rj 0 = a + a 2 , % = a -1 4- a -2 , or 7] a root of rf + r\ + 6 = 0 ; 
IV. Real numbers. 
I. ^) = 23m + l. First, it gives for the prime numbers of this form 47, 139,.., 967 
congruence roots, mod. p, and also congruence roots, mod. p 3 *; these last in the form 
a 4- bp + cp 2 , where a is given in the former table ; thus first table :— 
p a a 2 d 3 ... a 22 
47 6 - 11 - 19... + 8; 
and second table— 
P 
47 
CT 
13p - 23p 2 
+ p — 2p- + 13p — 23p 2 + 19p — 8p 2 ... 
The meaning is that, p = 47, the roots of the congruence 
a 22 + a 21 + ... + a 2 + a + 1 = 0 (mod. 47 2 ) 
a = 6 + p — 2p 2 , a 2 = — 11 + 13p — 23p 2 , &c. 
+ 22p 4- 22p 2 . 
are 
Secondly, it then gives f(a), the actual ideal prime factor of these same primes 
47, 139,.. , 967 ; viz. the whole of this portion of the table A = 23, I. (2) is, 
having actual prime factors, 
P /(«) 
599 a + — a 17 
691 a 3 + a 21 4- a 22 
829 a 2 4- a 5 4- a 46 ; 
having ideal factors, their third powers actual, 
P / 3 0) 
47 a 4 4- a 5 4- a 9 4- P 0 4- a 16 — a 20 + a 22 
139 1 — a 3 — a 7 4- a 9 4- a 11 4- a 14 4- a 13 4- a 17 4- a 18 4- a 20 4- a 21 
277 a 2 - a 4 - a 6 4- a 7 - a 10 - a 15 - a 17 4- a 21 4- a 22 
461 a — a 2 4- a 3 — a 1 ' + a 14 — 2a 13 
967 a 2 — a 3 — a 5 4- a 19 4- a 15 — 2a lfi 4- a 17 4- a 19 . 
I repeat the explanation that, for the number 47, this means /(a)/(a 2 ) .../(a 22 )= 47 3 . 
* 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.