700 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxui 



n = 5 (1, 1, 3, 3, 4), n = 7, 8, 10. He stated and L. de Jong proved that the 

 g.c.d. of solutions x, y, z of x 2 +y 2 +z z = xyz is 3, and listed seven sets of 

 solutions. Cf. Hurwitz 174 . 



G. Rados 194a proved that if a polynomial F(x) of degree n with integral 

 coefficients decomposes with respect to every prime modulus into n linear 

 factors with integral coefficients, then F(x) decomposes algebraically into 

 n linear factors with integral coefficients. 



A. Korselt 1946 argued that, if f(x, y} is a homogeneous function of degree 

 d>l with no multiple root, f(x, y)=z n is solvable in relatively prime 

 integral rational functions x, y, z of any parameters if and only if d=2, n 

 any, or d = 3, n = 2. 



" V. G. Tariste " stated and R. Goormaghtigh 195 proved ih&tx y y x =x y 

 has only the integral solutions x = y+l = l, 2, 3. 



M. Rignaux 1950 proved by the theory of quadratic forms that 



holds, when c= 1, only for K = 3. Cf. Hurwitz 174 . 



F. Irwin 1956 gave a method to find the integral solutions of 



ax r bxy -f y c = 0. 



For (x n 1)/(# 1) = D, see Landau, p. 57 of Vol. I of this History. 



On pr(p 2 -r 2 ) : gs(g 2 -s 2 ), see papers 67-77 of Ch. IV, Euler 81 of Ch. 

 XVI, Euler 18 - 19 of Ch. XVIII and Euler 253 of Ch. XXII. 



For k 2 +4kv= D, where k= (y+l)0 2 +l), see Haentzschel 144 of Ch. V. 



By Hilbert 54 of Ch. XIII an equation /=0 may have no rational solu- 

 tion, while /^O (mod p e ) is solvable when p is any prime. From one solu- 

 tion of F(x, y, 2)=0, Cauchy 150 of Ch. XIII found another. For 

 (/ 4 -& 4 )(0 4 -fc 4 ) = O, see Euler 28 and Gerardin 85 of Ch. XV, Ward 44 of Ch. 

 XIX. On /O) = D see Jacobi, 152 etc., of Ch. XXII. Brunei 68 of Ch. XXI 

 solved x n l +xl = F, where F is a cyclic determinant of order n. Euler 187 of 

 Ch. XXII noted rational solutions of abcd(a+b-{-c+d) = 1. 



MISCELLANEOUS SYSTEMS OF EQUATIONS OF DEGREE 



C. Gill and T. Beverley 196 found numbers whose sum is a 4nth power 

 and such that if the square of each be added to their sum there results a 

 square. Take px 2n , qx 2n , - as the numbers and x* n as their sum. The 

 final conditions give p 2 +l = D, <? 2 +l = D, r 2 +l = D, -, which hold if 



_y 2 -x 2n _ax 2n -y*fa _bx^-yr[b 



P 2yx n ' q= 2yx n 2yx n ~' 



To make p+gH ---- =x zn , take y=( a +b-\ ----- l)/(2z n ), l/a+l/6H ---- = 1. 

 J. Liouville 197 stated that, if there be a finite number of sets of positive 



ma Math, cs termcs. 6rtesito (Hungarian Acad. of Sc.), 35, 1917, 20-30. 



1946 Archiv Math. Phys., 27, 1918, 181-3. 



196 L'interm6diaire des math., 25, 1918, 30, 95. 



1360 Ibid., 131-2. 



1966 Amer Math Monthly, 26, 1919, 270-1. 



196 The Gentleman's Math. Companion, London, 5, No. 28, 1825, 367-9. 



197 Jour, de Math., (2), 4, 1859, 271-2. Cf. Gegenbauer. 2 '* 



