M.-P.— Vol. I.] DICKSON— THE ORTHOGONAL GROUP. 57 



being y 2 (fi n — e), where e = ± 1 according as fi n = 4 t ±1, 



the condition for O ' is that/" — e be divisible by 16. 



1 ,2 



A" 8 

 Since the condition that O be a Qi, 3 is that 2 (i±x) 



1 , 2 



be squares, the condition that O ' occur among the Q\, 2 



1 ,2 



is that 2 ± \/ 2 be squares. 



Corollary. 2+1/2 is a square in the G K \_p n ~\ only 

 when fi n is of the form 16 / ± 1. 



Thus ior p n ^= 17, X = 3, 2 -f- \/V=2 -f 6 = 5 2 (mod. 17) ; 

 for ^V 1 = 31, X = 4, 2 + 1/2 = 2 -f- 8 = ±4 2 (mod. 31). 



University of California, 

 April n, 1898. 



