CHAP, xxiii] SINGLE EQUATIONS OF DEGREE n>4. 693 



according as n is divisible by m or not, the number of roots of 0=0 (mod ra) is 





l\ 



, y , ...=o A-= 



When applied to (j> = x 2 +c, this formula leads to Gauss' results on trigo- 

 nometric sums. Again, x*+Ay-+B = Q (mod p) has p=tl sets of solutions. 

 Libri 148 noted that the number of sets of positive integral solutions of 

 <f>( x ) y, ) = an d the number of sets in which x, y, - - take the values 

 1, ,n l are approximately 



-"*', -"*' (-*+H-...), 



a;, y, ... = 1 r,y, ... = 1 



respectively. To apply the method of the preceding paper to the linear 

 congruence <t> = Ax 1=0 (mod p}, A not divisible by p, we use xv~ l 1=0, 

 or (Ax) p ~ l 1 . Since the division of the latter by < is exact, we get x = A p ~ z . 

 Next, for <j>=x~+qx+r=Q (mod 2p+l = prime), we divide x 2p 1 by < and 

 require that the remainder be divisible by 2p+l. Thus the conditions for 

 two roots a, |8 [neither zero] are 



1 _ a 2p-l \ 



- -1 + 1 = (mod2p+l), 

 a. / 



, 



p a p 



which by use of symmetric functions can be expressed in terms of q and r. 

 For the case x z s=0 (mod 2p+l), the first condition is satisfied and the 

 second reduces to s p 1=0. For z 2 -f-+l = (mod 6p+l), the first condi- 

 tion is equivalent to ( 3) 3p = l. 



V. Bouniakowsky 149 noted that there is an infinitude of solutions of 



x m X n +y m Y n =z m Z n , 



where m, n are relatively prime. Determine a, so that ma np = 1. Let 

 a and b be arbitrary and c = a-\-b. Then a solution is 



x = a", y = b a , z = c a , X = tfc f} , Y = aP<*, Z = a ft b . 

 New solutions follow from the integral form of 



a ma /a nf * +b m *'lb n *' = c ma "/c" ft ". 

 Similarly, if p, q, r, - are without a common factor, we may solve 



by use of 2L4;a; = 0, paq(3 - - = 1, replacing a { by af ?/3= "", throwing 

 negative powers into the denominator and clearing of fractions. 



G. C. Gerono 150 noted that if r is the radius of the circle inscribed in a 

 triangle with sides a, b, c and area A and if x = a/r, y = b/r, z = cfr, Heron's 

 formula for A, and A = | pr, where p is the perimeter, give 



(y+z-x}(x+z-y}(x+y-z') =4(x+y+z). 

 Call the factors 2X, 2Y, 2Z, respectively. Let x, y, z be positive integers. 



148 Mem. Aecad. Sc. di Torino, 28, 1824, 272-9; Jour, fur Math., 9, 1832, 59. 



149 Bull. Acad. Sc. St. Petersbourg, 6, 1848, 200-2. Cf . Hurwitz 212 of Ch. XXVI. 

 160 Nouv. Ann. Math., 17, 1858, 360. 



