CHAP, xvi] CONGRUENT NUMBERS. 465 



and 2rs = D . Thus x = 106921/D, z 2 + 13 = (127729/D) 2 , x* - 13 = (80929/D) 2 , 

 D = 19380. 



P. Barlow 40 proved by descent that 1 and 2 are not congruent numbers. 



J. Cunliffe 41 noted that if, when n is given, a rational v can be found for 

 which n+y 2 and n v 2 are rational squares, we can deduce a rational x 

 for which x~-\-n and x 2 n are rational squares. Take (a+6) 2 and (a b) z 

 as the latter. Then rc 2 = a 2 +6 2 , n = 2ab. To satisfy the former, take 

 a=(p 2 -q-)j(2r), b = pq(r. Then (p*-q~)pq = nr\ Takep = n,q = v 2 . Then 

 n 2 y 4 =D, which holds if nw 2 are squares. Application is made to the 

 case n = 13 by expressing 13 as a sum of two rational squares in two ways. 



11 Umbra " 42 noted that x 2 +n = a 2 , x z n = b- can be solved if 



n=(c 2 -M 2 )/s 2 , c 2 -d 2 =D. 



For, 2z 2 = a 2 +& 2 is known to hold if a, b = (2pqp 2 =Fq z )lr, x = (p z +q")/r. 

 Then n = \ (a 2 6 2 ) = 4pq(p z q^/r 2 . Taking p = c 2 , q = d 2 , we have 



r 2 = 4cW(c 2 -d 2 ), 



whence r is rational since c 2 d 2 = D. Similarly, z 2 w= D are solvable if 

 tt = (c 2 d 2 )/s 2 , c 2 +d 2 =D, or if n is double the sum of two squares the 

 double of whose difference is a square. 



A. Genocchi 43 noted that the problem to make x-hq 2 both squares is 

 equivalent to the single equation x 4 Wq* = D . By the direct, but laborious, 

 method of Fermat (on Diophantus VI, 26), used by Lagrange (see papers 

 37-41, 54 of Ch. XXII), Genocchi treated the example h = 5 far enough 

 to reach the special solution due to Leonardo. 4 The direct solution of 

 z 2 /i=D leads to 4mn(m z n-) = hg 2 or the problem to form a rational 

 right triangle with a given area. The absence of a treatment of the latter 

 leaves an evident lacuna in Diophantus VI, 6-11 (V, 8 deduced a new solu- 

 tion from one). The method by Euler 33 is identical with that of Leonardo. 



Genocchi (pp. 206-9) proved that an integer y is of the form 4ran(ra 2 n 2 ) 

 in only a finite number of ways. To two solutions x of x-y= D, each x 

 a sum of two squares, correspond distinct values of y. From one solution 

 (pp. 251-3) of z 2 zbfc= D, we readily get others. Cf. Young 134 of Ch. XIX. 



Genocchi 44 proved that r 4 +4s 4 , 2r 4 +2s 4 , r 4 -s 4 are congruent numbers; 

 also r 4 +6r 2 s 2 +s 4 and db(r 4 -6r 2 s 2 +s 4 ) if one of the integers r, s is even and 

 the other odd. No prime 8m +3 is a congruent number. 



Genocchi 45 proved that the double of a prime 8fc+5 is not a congruent 

 number. 



Matthew Collins 46 proved that the only congruent numbers <20 are 

 5, 6, 7, 13, 14, 15; that a prime a = 4n+3 is not a congruent number if, 



40 Theory of Numbers, London, 1811, 109, 114. 



41 T. Leybourn's Math. Quest, from Ladies' Diary, 3, 1817, 368-71. 



42 The Gentleman's Math. Companion, London, 4, No. 21, 1818, 750-2. 



43 Annali di Sc. Mat. e Fis., 6, 1855, 129-134, 291-2. 



44 Ibid., 313-7. Cf. Genocchi. 68 



45 II Cimento, Rivista di Sc. Let ed Arti, Torino, 6, 1855, 677-9. Genocchi, 7 p. 299 for the 



number 10. 

 46 A Tract on the possible and impossible cases of quadratic duplicate equalities . . . , Dublin, 



1858, 60 pp. Abstr. in British Assoc. Reports for 1855, 1856, II, 2-5; and in The Lady'a 



and Gentleman's Diary, London, 1857, 92-6. 

 31 



