594 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



G. Eisenstein 314 proved that, if p is a prime 3m +1, 27(x p ~ l -\ hc+1) 



can be expressed in the form 



$ = u? + pp i?/ 3 + pp 2 z* - Spuyz, 



where y = v+wp, z = v-\-wp z , and u, v, w are polynomials in x with real 

 coefficients, while p 2 +p-fl = 0, and pi, p 2 are the primary complex prime 

 factors of p. The product of two forms 4> is of like form. When <t> = l 

 has real integral solutions other than u = l, y = z = Q, an infinitude of solu- 

 tions can be derived from one, as by Pell's equation. 



C. Souillart 315 and E. Mathieu 315 proved that the product of two forms 



x y z 



y z x 



z x y 



is of the same form and stated that a like theorem holds for cyclic deter- 

 minants of order n. This was proved for C by J. Petersen. 316 



E. Meissel 317 wrote (x, y, z) for x*+Ay 3 +A 2 z 3 SAxyz, where A is posi- 

 tive and not a cube. Let 3 = 1 and x, y, z be integral solutions of 

 (x,y,z) = l. Let 



(re + 6yp + 2 zp 2 ) (a + dbp -\- 2 cp 2 ) = 1 , p = VA . 



By the product of this for the three values of 0, we get (x, y, 2) (a, b, c) = 1. 

 By the three equations which follow from the above, 



a = x z Ayz, b = Az~xy, c = y-xz, 



which give a second solution of (x, y,z) = l. An nth solution follows from 

 (z+0i/p-l-0 2 zp 2 )". Solutions of (x, y, z) = l are found for each A<82. 

 G. B. Mathews 318 proved that if the integer m can be represented by 



F(x, y, z} x^+ny^+rfiz* Snxyz, 



it can be represented in an infinity of ways. F(x, y, z) = 1 has integral 

 solutions and all solutions can be derived from a single fundamental solution 

 , 77, by use of 



H. W. Lloyd Tanner 319 wrote tf>(x, y, z) for the norm of x+y6+ze*, 

 where 3 +3A;0 6 = 0, and called u+v6+w6- a unit if <t>(u, v, w) = l. He 

 obtained a correspondence between the units and the proper automorphs 

 of <f>, i. e., linear transformations of into itself, and investigated improper 

 and associated automorphs. 



H. S. Vandiver 320 noted that the circulant (cyclic determinant) of order 

 n is a product of n linear factors 



+ co n -*a n (fc = 0, 1, , n 1), 



314 Jour, fur Math., 28, 1844, 289-303. 



316 Nouv. Ann. Math., 17, 1858, 192-4; 19, 1860, 320-2. Cf. Math. Quest. Educ. Times, 63, 



1895, 35-C. 



819 Tidsskrif t for Math., 1872, 57. 



817 Beitrag zur Pell'schen Gleichung hoherer Grade, Progr., Kiel, 1891. 

 "" Proc. London Math. Soc., 21, 1891, 280-7. On F = 0, see Maillet 160 of Ch. XXIII. 

 " 9 Iliiil., 27, 1895-6, 187-199. 

 M0 Amer. Math. Monthly, 9, 1902, 96-8. 



