CHAP, xvi] CONGRUENT NUMBERS. 4G7 



Like results hold if we take (cf. Lucas nl ) 



g r 4 _ 2Qr 3 - 2r 2 +20r +9 = w\ 



2(2r+l) 



P r> / t \ fc\ i i \ > o. 



If &/ is a congruent number and hence equal to 4a/3(cr /3 2 )p 2 /<? 2 , we may set 

 & = 2X (2a/3) , K ' = X (a 2 - /3 2 ) , X' E= Ktfq'-fp*. 



Thus if also K and hence K' is a congruent number, then k is the double 

 of a leg of a right triangle whose second leg is a congruent number. 



If kK = Ki is a relation between three congruent numbers, the last 

 formulas show that a = 2X/3 and <TI = 2Xa are solutions of the system 



where 4> = 4\K f , ^ = Xfc. Conversely, if one of these equations can be 

 solved, kK' and hence kK is a congruent number. 



To find congruent numbers K, K such that kK = KI, where & is a given 

 congruent number, take as KI in turn the 12 types in the earlier paper, 47 

 each type multiplied by an arbitrary rational square. Give KI the form 

 4aj3(a 2 /3 2 )p 2 /# 2 , and equate the latter to kK. Hence 



so that the leg a 2 (3- of a rational triangle is a congruent number and the 

 other leg 2a/3 is k/2. But this solves kK = Kt for K. 



G. Le Secq. Destournelles 49 proved the impossibility in integers of the 

 pair 



z 2 +?/ 2 = 2 2 , X'-y^u^. 



The equation obtained by adding these may be written 



V 2 / 2 /' 



The terms on the right may be assumed relatively prime. Thus 



z+u z u a- jS 2 



-y-=/3, ^ = ~2-' 



or vice versa, where a, are odd relatively prime integers. Substituting 

 either set into 2y- = z 2 u-, we get 



2/ 2 = l8(a 2 -j8 2 ), a = w 2 , /5 = n 2 , m 4 -n 4 =D. 

 Thus m 2 n 2 = 2^ 2 , 2i 2 , so that 



But these are like the initial equations with /j<o:, l<y. 



A. Genocchi 50 stated that x-h are not both rational squares when h 

 is a prime 8ra+3 or the product of two such primes, or the double of a 

 prime 8m +5, or the double of the product of two such primes. 



49 Congrgs Sc. de France, Rodez, 40, I, 1874, 167-182; Jornal de Math, e Ast., 3, 1881. 

 60 Comptes Rendus Paris, 78, 1874, 433-5. Reprinted, Sphinx-Oedipe, 4, 1909, 161-3. 



