CHAP, xvi] CONGRUENT NUMBERS. 471 



G. Bisconcini 64 determined the numbers A=4rs(r 2 s 2 ) which are 

 products of powers of three primes. If 2 and 3 are the only prime factors 

 of A, then A = 24. 



R. D. Carmichael 6 ' 5 proved that the system q 2 -\-n 2 = m 2 , m 2 +n 2 = p 2 has 

 no positive integral solutions [whence w 2 +n 2 = p 2 , m i n" = q^~\. 



H. B. Mathieu 66 asked if x 2 +A=u 2 , x 2 A = v 2 are completely solved 

 by the identity 



L. Aubry 67 replied that all solutions of 2x 2 = u 2 -\-v~ are given by 



u, v = l(r2rs-s 2 )', x = l(r"+s z ); A=4l 2 rs(r 2 -s 2 ). 

 The case / = !, r=afr, s = b, gives the above identity, which with 



give all relatively prime solutions. [Cf. Ch. XIV 



G. Metrod 68 treated x 2 +y = u 2 , x 2 -y = v 2 . For w x = a 2 -6 2 , etc., 



Hence (u l +v 1 ) 2 +(u 1 -v l y = 2x 2 ; u, v = a 2 -b 2 2ab, y = 4ab(a 2 -b 2 ). 



J. Maurin and A. Cunningham 69 noted that from one solution of 

 x 2 ny 2 = z z , x 2 -{-ny 2 = t 2 , we get a second solution X = x*+ri 2 y 4 , Y = 2xyzt. 



A. Gerardin 70 listed the values <1000 of h for which z 2 % 2 =D for 

 x < 3722. He noted (pp. 57-9) the solutions 



x= / 8 + 6/V+flf 8 , 2/ 



L. Bastien 71 listed the 25 values <100 of h for which x 2 /M/ 2 are not 

 squares, and stated (besides Genocchi's 50 results) the following cases of im- 

 possibility: h the double of a prime 16m+9; h a prime Sm-\-l = g 2 +k 2 , 

 with gr+A; a quadratic non-residue of h (as for 17, 73, 89, 97). 



T. Ono 72 noted that x 2 5y 2 are squares for z = 41, y = l2 [Leonardo 4 ] 

 and x = 3344161, y = 1494696. 



G. Candido 73 noted that, from two sets of solutions (#,-, yi) of the system 

 x 2 wf-= D, we get a third set by Euler's identity 



(X 1 x 2 uy 1 y 2 ) 2 +u(x l y 2 :: fuy 1 x 2 ) 2 = (x\+uy\] (xl+uyl). 



E. Turriere 74 noted that if x, y, z are the rational coordinates of a point 

 M on the quartic space curve x 2 -\-a = y~, x 2 -\-b=z 2 , the osculating plane at 

 M is 



Periodico di Mat,, 24, 1909, 157-170. 



65 Amer. Math. Monthly, 20, 1913, 213-6. 



66 L'interm&iiaire des math., 20, 1913, 2. 



67 Ibid., 211-2. Practically same by Welsch, 212-3. 



68 Sphinx-Oedipe, 8, 1913, 130-1. 



69 L'interm&liaire des math., 21, 1914, 20-21, 176-8. 



70 Ibid., 22, 1915, 52-3. 



71 Ibid., 231-2. 

 Ibid., 117. 



73 Ibid., 23, 1916, 111-2. 



74 L'enseignement math., 17, 1915, 315-324. 



