CHAP, xvi] CONCORDANT FORMS. 477 



G. C. Gerono 89 took t = l, without loss of generality. Since u is odd, 

 u = 2k+l, the first condition gives w 2 = fc 2 +(/+ 1) 2 . He proved that also 

 ?' = ra 2 -j-(??i+l) 2 . Using the [unproved] theorem of de Jonquieres 26 of 

 Ch. XVII, we get v = 5 (excluded) or 1. 



T. Pepin 90 noted that Lucas 88 did not treat all possible cases, whereas 

 the omitted cases add new solutions. By using a somewhat different 

 method, we get all solutions by a single set of formulas. We may limit to 

 relatively prime solutions t, u and take v and w positive. By the first 

 equation, 



for a, 6 relatively prime, a odd. By the second given condition, 



Comparing the values of t and of u, we get equations equivalent to 



Thus a = a\, 6 = 3/fy, c = a/3, d = 4X/x. Inserting these into the difference of 

 the two preceding equations, we get a quadratic giving 



V(3/3 2 -4X 2 )(2X 2 -3/3 2 j 



a~ 2(9/3 2 -8X 2 ) 



Since the radical must be rational, 3/3 2 -4X 2 =7 2 , 2X 2 - 3/3 2 = 5 2 . The 

 upper signs are excluded modulo 3. Hence 2X 2 = 7 2 +5 2 , 3/3 2 = 7 2 +25 2 , 

 a pair like the given pair. Hence a solution v, w, t, u leads to a second solu- 

 tion 



where n :a = vwtu : 2(9w 2 8y 2 ). Starting from v = w = t = u = l, we get 

 ju/a = or 1, the second giving W! = 37, Wi = 33, ti = 47, ^i = 23; etc. It is 

 proved that we get all solutions in this way. 



To find 91 two squares whose sum is double a square and difference is 10 

 times a square, take x, y = 2pq(p" q~'). Then x 2 +2/ 2 = 2(p 2 +g 2 ) 2 , 

 x z -y z =8pq(p 2 -q*) = W(12m?) 2 if p = 5m, g = 4ra. 



J. H. Drummond and W. F. King 92 proved that 2x"-y-= D, 2if-x-= D 

 imply x- = y 2 . 



A. Gerardin 93 noted that x 2 -\-ny 2 and nx-+y z are squares if 



n=( 2 +/3 2 ) 2 -!, 



or n = 7, = 3, y = l, or n = l7, x = 8, y = l. 



Several writers 94 gave solutions of the last problem. 



R. Goormaghtigh 95 made Sx 2 +Py 2 and Sy-+Px- both squares. 



L. Aubry 95a proved that 2y z -{-u~ and 3?/ 2 +w 2 are not both squares. 



89 Nouv. Ann. Math., (2), 17, 1878, 381-3. 



90 Atti Accad. Pont. Nuovi Lincei, 32, 1S78-9, 281-292. 



91 Math. Quest. Educ. Times, 63, 1895, G4. 



92 Amer. Math. Monthly, 6, 1899, 47-8, 151-5. 



93 L'interme'diaire des math., 22, 1915, 128. 



94 Ibid., 23, 1916, 63-4, 205-7. 



95 Ibid., 184-5. 



950 Ibid., 26, 1919, 84-5. For u = 1, Rignaux. 131 



