48 CHAPTER IV. 



66. Theorem. The number of sets of solutions in the 

 of the equation . ,2 , fc 8 , 



where each ccj is a mark ={= in the field and K belongs to the field, 

 isp 2nm + op nm , where = 4- 1, 1 or according as ( 

 is a square, a not -square or zero in the field. 



Consider the equivalent system of equations 



' ' 



The first equation has one solution if y = 0. If ?/ =J= 0, it has two 

 or no solutions according as a r] is a square or a not -square. Let 

 u, = if x ? and /u, = 1 according as a^x is a square or a not- 

 square. We may express the number of solutions of the second 

 equation by 65, if we set v = 1 according as ( I) w a 2 . . . 2 m+i 

 is a square or a not- square. Evidently we have (JLV = co. 



According as /i = 0, + 1, or 1, the total number of sets of 

 solutions of the pair of equations is respectively 



. ) 



-Ili J|jj(2 1)_ 



In each of the three cases, we have enumerated separately the number 

 of solutions arising when iq = 7 when 17 == ^ and when 77 is one of 

 the values =j= for which the first equation has solutions (viz., two). 



67. Theorem. - - If S denote the number of squares 1 ) e 2 in the 

 GF[jp n '\ for ^vhich 6 2 -f 1 is a square and N the number of square r 2 

 for which T 2 + 1 is a not -square, we have 



S=jGP"-5), JV-j(p-l), if -1 -square; 

 S - \(P H -3), N = i(p + 1), if - 1 = not-square. 



Indeed, the number of sets of solutions jj, y in the (^^[^ re ] of the 

 equation 2 = ^2 _|_ | 



is always p n 1 (by 64). These solutions are of three kinds: 



1. 6-0, *-l 5 

 2. | 2 = - 1, , - 0, 

 occurring when 1 is a square; 



3. ^=4=o, ^ 2 = + i4=o ; 



giving 4S sets of solutions , ^. 



1) The mark zero is not reckoned as a square. 



