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



for m<a/2, m 2 2 is not divisible by a (examples: a = 11, 19, 43). To 

 treat x 2 5y 2 =n, we add and subtract and get 2x 2 =z 2 +w 2 , Wy 2 =z 2 w 2 . 

 Set z = z'-{-w r , w=z' w r , where z', w' are relatively prime. Thus 



whence z' = m 2 n 2 , w' = 2mn, x = m 2 -\-n 2 , and z'w' = 5y~/2, whence y = 2y', 

 mn(m 2 n 2 } = 5y' 2 . If n is divisible by 5, n = 5g 2 , m = p 2 , ra-fn = r 2 , 

 mn = s 2 , leading to a pair like the initial equations, so that this case is 

 excluded. If m = 5p 2 , we get n = q 2 , mn = r 2 , s 2 , whence 5p 2 +# 2 = r 2 , 

 5p 2 q 2 = s 2 . As the latter are satisfied by p = l, q = 2, whence m = 5, ft = 4, 

 we get the solution [Leonardo's] # = 41, y = l2, z = 49, w = 31. In general, 

 given a solution of 



ax 2 -\-by~ = nz 2 , dbx 2 ?/ 2 = 

 then X = n0 4 +w 4 )/2, F = 2z7/zw; make 



and hence give a solution of X 2 +abY 2 = D, X" 2 a6F 2 = D. For example, 

 if a = 5, 6 = n = l, we have 5# 2 ?/ 2 =n, holding for x = l, y 2, whence 

 X = 41, 7 = 12 satisfy X 2 5F 2 =D. 



F. Woepcke 47 found 12 congruent numbers associated with the given 

 one 2xy, where x*-\-y 2 = z~, viz., zx, zy, x 2 y 2 , z 2 +x' 2 , z-+y 2 , 4xy(x 2 y 2 ), 



( 2 _|_ rr \2 

 -~- J 



In fact, x = a 2 b 2 , y = 2ab. In 2xy replace a by z and b by x and drop the 

 square factor 4(z 2 x 2 ) =4i/ 2 ; we get xz = a 4 6 4 . But if we replace 6 by y, 

 we get ?/2. In a 4 fe 4 , take a = x,~b = y, and drop the square factor x 2 +2/ 2 = 2 2 ; 

 we get x 2 y 2 . Double the product of the latter by the congruent number 

 2xy is a congruent number; etc. He computed the above 12 functions for 

 each right triangle in the Arab manuscript. 1 



Woepcke 48 treated the problem, proposed to him by Boncompagni: 

 Given a congruent number k, to find a congruent number K such that 

 the product kK of the two is another congruent number. If k is formed from 

 a, 6, where 2a 2 6 2 = c 2 , then 



a6(a 2 -6 2 )-ac(a 2 -c 2 )=6c-a 2 (6 2 c--a 4 ). 

 If k is formed from two numbers of ratio r, where 



and K is formed from two numbers of the ratio 



-(r l) 2 d=w 

 P ~ 2(r-l) ' 



then kK is a congruent number formed from two numbers of the ratio 

 0- = (-r 2 +3w>)/2. For, then 



/ 1W 1\ 1 p q 1 



(r- )(p } = ff , ! -^j-ri 



\ r/\ p/ a q p 4p~q 2 



47 Annali di Mat., 3, 1860, 206-15. Same in Atti Accad. Pont. Nuovi Lincei, 14, 1860-1. 



259-67. 

 " Annali di Mat., 4, 1861, 247-55. 



