CHAP, xvi] CONGRUENT NUMBERS. 469 



Lucas 53 noted that if there exist relatively prime solutions of 



x 2 Ay 2 = u-, x-+Ay 2 = v 2 , 

 then A is of the form Xjii(X 2 ju 2 ). For, by addition, 



(U + V\ (U-V\ U + V U-V 



\~2~) + V~2~J =X > -2- = *-r> "2~ = 2X ^ * = *-+M 2 , 

 where X, n are relatively prime and one is even. Hence 



u, y = X 2 -M 2 2X/z, 7/ = 2, A = X]u(X 2 -M 2 ). 



He next showed how to derive a second solution from one, given that 

 A is a congruent number resolved into its prime factors. If a, ft are two 

 integers whose product is A, the second equation gives 



v+x = 2ae 2 , v-x = 2pf 2 , y = 2ef. 

 Substitute the resulting v, x into the first given equation; then 



The two factors of the left member equal 2/3?gr 4 , 4/3l/i 4 , where Pif3 2 = (3, 

 gh =/. For the upper sign, we add and get 



The two factors equal atf 2 and 2 g 2 , where ai<x z = <x, pq = e. Taking 

 i = ce 2 = /3i = l, we have a system like the proposed. Hence a solution 

 x, y, u, v leads to the second solution 



For the lower sign above, we obtain a complicated set of formulas giving 

 a new solution from one. The formulas are said to give all solutions when 

 A = 6 and for the problem x 2 (x+2) = D of Beha-Eddin. 32 By combining 

 Lucas' result (ibid., p. 433) with the results of Fermat and Genocchi, 45 

 Lucas concluded (p. 514) that xy(x 2 y 2 )=Az 2 has no rational solution if 

 A = l, 2, p, 2q, where p and q are primes of the respective forms 8n-f-3, 

 8n+5. 



S. Giinther 54 treated x 2 +a = ?/ 2 , x~ a = z L by setting 



which determine y and z in terms of x and m. Substituting these into one 

 of the proposed equations, we get x as a function of m: 



4w 4m 3 



Set m ap 2 . Then x is rational if 1 a?p 4 D . Hence we seek among the 

 rational solutions of 1 a 2 2 =7? 2 those values of which are squares. If 

 such exist, a is a congruent number, otherwise not. We can not go further 

 with the general solution of the system since the character of a decides 

 whether or not such a biquadratic root of the Pell equation exists. 



M Nouv. Ann. Math., (2), 17, 1878, 446. 

 " Prag Sitzungeberichte, 1878, 289-94. 



