468 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvi 



E. Lucas 51 noted that a is a congruent number if and only if x 4 a~y* = z* 

 is solvable; then a = 0, 1 (mod 5) if xy is not divisible by 5. A congruent 

 number does not end in 2, 3, 7 or 8 when y is not divisible by 5. We are 

 led to congruent numbers by the problem to find three squares in arith- 

 metical progression whose common difference is the product of a by a 

 square. The equations (pp. 184-6) 



(4) z 2 -5i/ 2 = w 2 , z 2 +5?/ 2 = y 2 



were studied, but not completely solved, by Leonardo, 4 - 16 Paciuolo, 20 Euler, 33 

 Collins, 46 and Genocchi, 7 p. 289. We may assume that x, y, u, v are rela- 

 tively prime, so that x and v are odd, y even. Hence in view of the first 

 equation we may set 



x w = 1 Or 2 , x+u = 2s 2 , y = 2rs (r, s relatively prime) . 

 By the second equation (4), (5r 2 +3s 2 ) 2 8s 4 = y 2 , whence 



5r 2 +3s 2 v = 2p 4 , 5r 2 +3s 2 Ty = 4g 4 , s = pq. 

 Adding the first two of these we get 



Since the factors on the left are relatively prime, we find after considering 

 residues modulo 5 that the only two admissible cases are p 2 g 2 = 5</ 2 , 

 p 2 2<2 2 ==b/i 2 . For the upper sign, the evident solution p = 3, <? = 2, 

 g = h = l, leads to Leonardo's solution x = 41, ?/ = 12, it = 31, y = 49 of (4). 

 For the lower sign, we get the system q- 5g 2 = p 2 , q--\-5g <2 = h~, like (4); 

 hence from one solution we get the second : 



X = u z x 2 +5vY, U = u 2 x 2 -5v 2 y 2 , V = u 4 -2x\ Y = 2xyuv, 



which differ only in form from the formulas by Genocchi. Lucas solved 

 (pp. 191-3) the equation to which Woepcke 48 was led: 



9a 4 -20a 3 6-2a 2 5 2 +20a6 3 +96 4 = c 2 . 



This may be written d 2 +44a 2 6 2 = 9c 2 , where d = 9a 2 -10a&-96 2 . Thus 



3cd = 2p 2 , 3cTd = 22g 2 , ab = pq. 



Set b = mq, p = ma. From p 2 Ilg 2 =d we get a quadratic for m with 

 a rational root if 13aV(a 4 4-ll# 4 ) =z 2 . For the upper sign, 



(2a 2 +13g 2 ) 2 -42 2 =125g 4 . 

 If we take the factors of the left member to be r 4 and 125s 4 , and add, we get 



(r 2 -13s 2 ) 2 -4a 2 = 44s 4 . 

 Call the factors of the left member =t2w 4 , T22t> 4 ; adding, we get 



which is like the initial quartic, but with smaller values of the unknowns. 

 A like result is proved in the remaining admissible cases. The system 62 

 z 2 6?/ 2 = D is treated (pp. 180-4) by the method used for the generalization 

 given in the next paper. From Leonardo's solution x = 5, y = 2, is deduced 

 1201 2 6-140 2 = 



Bull. Bibl. Storia Sc. Mat., 10, 1877, 170-193. 

 "Also in Nouv. Ann. Math., (2), 15, 1876, 466-70. 



