290 
ME. n. T. EENTsM]TT ON THE EESIDUES OF POWEBS OF NUMBERS 
APPENDIX. 
Tables of Indices for all Moduli whose Norms do not exceed 100. 
Description of Tables. 
The following Appendix contains Tables of Indices for all the moduli whose norms 
do not exceed 100. For each modulus two tables are given ; the first arranged so as 
to show readily the number that corresponds to given indices, the second to show 
what indices correspond to a given number. At the foot of any column, or the end 
of any row in which indices are tabulated, is placed in a bracket the exponent of the 
generator to which those indices refer. With each table are noted the formulae 
necessary for finding to which of the numbers in the table any given number is 
congruent. There are also given for convenience the prime factors of the modulus, 
the norm, the highest exponent, the value of expressed in factors which show the 
exponents of the generators, and the generators used in the table. All these will be 
found collected in the reference table next following. In this are also noted, for 
each modulus, the least possible number of generators and the values of the numbers 
^ of Proposition xix. E.g., for tlie modulus 5 + iyi we read thus ;— 
5 + 5i= -?■(!+0(2 + 0(l+ 2t), N(5 + 5t)=50, (5 + 5^) = 16 = 22 21 
Highest exponent = 4. 
being 4 -f t and 9 + 
Also if 
then 
Least number of generators = 2, those used in the table 
a = olq (mod 1 + 'i) 
= (mod 2 -f- i) 
= ol .2 (mod 1 fi- 2i) 
a — Sag -b (3 + i) «! + (8 + 4^) (mod 5 + 5i). 
The reducing formulae (see preface, Part II.) are 
y = T (mod 5), 
.T = X + Y — y (mod 10). 
The tables of indices for powers of 1 + i as moduli, up to (1 + t)®, are placed at 
the end. 
[Tlie tables have been calculated with some care, but they liave not been revised.] 
