86 
SPECIMEN TABLE M = a a ¥ (mod. N) &c. 
[397 
[from p. 83] of these is in fact 1, 3, 7, 9, 11, 13, 17, 19, each of which is expressible 
in the required form, viz. 1 = 3°. 11°, 3 = 3*. 11°, 7 = 3 3 .11°, &c. (Mod. 20): the maximum 
indicator is 4; viz. 1 4 =1, 3 4 = 1, 7 4 = 1, &c. (Mod. 20). 
The table pp. 84, 85 gives the Indices for the numbers less than N and prime 
to it, for all values of N from 1 to 50; the arrangement may be seen at a glance; 
of the five lines which form a heading, the first contains the numbers N; the second 
the root or roots belonging to each number N, the third the indicators of these roots, 
the fourth the maximum indicator, the fifth the number <£ (N). The remaining lines 
contain the index or indices of each of the cf>N numbers M less than N and prime 
to it, the number corresponding to such index or indices, being placed outside in the 
same horizontal line. For example, 30 has the roots 7, 11, indices 4, 2 respectively; 
the Maximum Indicator is 4, and the number of integers less than 30 and prime to 
it is 8; taking any such number, say 17, the indices are 1, 1, that is, we have 
17 = 7 1 . II 1 (Mod. 30). 
The foregoing corresponds to the Tabulae Indicum Numero dato cor respondentium of 
Jacobi; on account of multiplicity of roots there does not appear to be any mode of 
forming a single table corresponding to the Tabulae Numerorum ad, Indices datos perti 
nentium ; and there would be no adequate advantage in forming for each number N 
a separate table in some such form as 
N= 20. 
Roots 
3 11 
Nos. 
0 
0 
1 
0 
1 
11 
1 
0 
3 
1 
1 
13 
2 
0 
9 
2 
1 
19 
3 
0 
7 
3 
1 
17 
which I have written down in the form of a table of single entry; for although 
(whenever, as in the present case, the number of roots is only two) it might have 
been better exhibited as a table of double entry, when the number of roots is three or 
more it could not of course be exhibited as a table of corresponding multiple entry.