678 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxm 



has an infinity of integral solutions. Application is made to functions 

 considered by Lagrange 142 which repeat under multiplication. If such a 

 function can take a given value, it takes the same value for an infinitude of 

 sets of values of #, , z, under the assumption that the algebraic equation 

 to which the function owes its origin has no rational divisor, but has at 

 least one real root. 



G. Libri 20 stated that the conditions imposed on (1) that there be a 

 real root and no rational factor are not necessary, it sufficing to have h = 1. 



J. Liouville 21 proved Libri's theorem false. For, if (1) is s 2 +l = 0, 

 then (2) is (x-\-yi)(xyi) = x z -\-y' 2 =l, with only a finite number of integral 

 solutions. 



Dirichlet 22 noted that his theorem remains true if (1) has only imaginary 

 roots, provided n>2. The problem is that of the units of an algebraic 

 domain. 



P. Bachmann 23 treated the solution of N = 1, where N is the norm of the 

 general algebraic number determined by a root of an equation of degree n. 



H. Poincare 24 noted that, for F defined by (2) by means of any equation 

 (1), the problem to find integers /?, such that F(/3i, , /3 n ) shall equal any 

 given integer N reduces to the problem to form all complex ideals of norm N. 

 In the solution of the latter one considers the congruences s n +as n ~ 1 + =0 

 (mod //), fji any divisor of N. 



E. Meissel 25 considered the product, extended over the roots of 5 = 1, 



7= (x, y, z, u, v)=IL(x+eyp+e 2 zp <2 +8 s up*+6*vp*), P = VJ. 



By the reciprocal solution of V=l is meant l/V=(a, b, c, d, e) = l, where 



dV 3V dV 3V 3V 



5Ac = -, 



. , , , . 



dx dy dz du dv 



For 2^A^7, he gave two primary solutions Vi = l, F 2 = l, accompanied 

 by their reciprocal solutions. He stated that two primary solutions always 

 exist and deduced the solutions F^Fz. He conjectured that, if p is a 

 prime, the corresponding Pell equation of degree p has f(p 1) primary 

 solutions. 



A. Thue 26 considered a homogeneous polynomial F(XI, - - , x n ) of degree 

 n 1 such that F=Q can be given the form 



(3) PlPz "Pn-l QlQz ' Qn-l, 



where Pi, Q { are linear functions of x\, , x n with integral coefficients. Set 

 (4) aiPi = a 2 Qi, aiP^asQz, , a n -\P n -i = a\Qn-i, 



where the a's are any integers without common divisor. Then (4) if 

 independent give #; = fcA t - (i = l, -, ri), where A,- is a homogeneous poly- 



20 Comptes Rendus Paris, 10, 1840, 311-4, 383. 



21 Ibid., 381-2. Bull, des Sc. Math., (2), 32, I, 1908, 48-55. 



i2 Bericht Akad. Wiss. Berlin, 1842, 95; 1846, 103-7; Werke, I, 638-644. 



23 De unitatum complexarum theoria., Diss., Berlin, 1864. 



24 Comptes Rendus Paris, 92, 1881, 777-9; Bull. Soc. Math. France, 13, 1885, 162-194. 

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



28 Det Kgl. Norske Videnskabers Selskabs Skrifter, 1896, No. 7 (German). 



