CHAP, xvi] CONCORDANT FORMS. 475 



To avoid fractions set a = 2; then y- = 4(m+ri)(r- v?). Set m+n=fg, 

 l- u 2 = 4fgz 2 . Hence t=fz+g, u=fz~ g, y = 4fgz, and we obtain "our 

 fundamental solution ' 



Let k- be the maximum square dividing mn = k 2 L, and set 



Then 2 +L = A 2 . Now fg2kA must be a square fz-, say B 2 . Thus 

 y = 4gB. Hence to exhibit the simplest solution, set 83 m-\-n=fg = k-L, and 

 let 4> be any factor of L such that 2 +L = A 2 is satisfied [identically] by 



Let 5 = fcC. Then fg2kA = 2 becomes 



If the latter can be solved, we have x = (/< $)/2, ?/ = C, since z : ?/ = : C. 

 It is proved (p. 33) that x 2 +my 2 and x~-\-ny 2 are concordant if 



since they equal (mw+2m+l) 2 and (wn+2n+l) 2 for x = mn l, y = 2P; 

 also, if m-f-ft = 2Q 2 , since they equal (3m+n) 2 and (3n+ra) 2 for x = m n, 



M. Collins 46 proved that a; 2 +?/ 2 =n, a; 2 +a?/ 2 =n are impossible for 

 lz<20, except for a = 7, 10, 11, 17; also, o; 2 -?/ 2 =n, a; 2 -a?/=n for 

 Ka<13, except a = 7, 11. If we know solutions of 



x 2 + ay 2 = n2 2 , y 2 +bx 2 

 then X = x' 2 w z y~z z and F = 22/2W are solutions of 



C. H. Brooks and S. Watson 84 found that x*+y" and x-+Ay 2 can be 

 simultaneously squares only for the following 41 positive integers A ^100: 

 1, 7, 10, 11, 17, 20, 22, 23, 24, 27, 30, 31, 34, 41, 42, 45, 49, 50, 52, 57, 58, 

 59, 60, 61, 68, 71, 72, 74, 76, 77, 79, 82, 85, 86, 90, 92, 93, 94, 97, 99, 100. 

 Set xfy = v, v 2 +l = 0+n) 2 , v 2 +A = (v pri) 2 . The two rational values of v 

 give n 2 = (A+p)/(p 2 +p), where p may be any positive or negative integer 

 or fraction for which A is integral. 



S. Bills 85 gave a theorem said to include all the theorems by Collins. 46 

 The equations x*+Ay*= D, x-+By 2 = D, cited as (F), are satisfied if 



A B 



x = mp 2 -- <7 2 = nr 2 -- s 2 , y = 2pq = 2rs. 

 m n 



In view of the latter, take p =fg, q = hk,r =fh, s = gk. Then the former holds 



83 But the author had previously set mn=k 2 L. 



84 The Lady's and Gentleman's Diary, London, 1857, 61-3, Quest. 1911. 



85 The Lady's and Gentleman's Diary, 1861, 82-4. 



