456 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xv 



C. Gill 64 found five numbers the sum of every three being a square. He 

 used trigonometry. 



To find three integers in geometrical progression, such that each plus 

 unity is a square, Judge Scott 65 took x 2 1, 2x(x 2 1), 4x-(x 2 1). It remains 

 only to satisfy 2x(x 2 !) + ! = D =p 2 ; take 2x-\-2 = pl, x 2 x = pTl. 

 A. Martin used x, xy, xy 2 and took y = a?x-\-2a. Then xy + l = [3, 

 xy 2 +l = (l+2a 2 x) 2 if z=(4a-4)/a, and x+l = b 2 gives a. D. S. Hart 

 used x, xy, xy 2 with x = m 2 -}-2m. 



A. Emmerich, 66 to solve 4+5 = w 2 , 5x-\-4 = v 2 , eliminated x to show 

 that u = 3a, f = 3/3, 5a 2 4/3 2 =l, every solution of which is given by 



2/3o:V5=(2V5) 2n+1 . 



To find 67 three integers in arithmetical progression such that the sum 

 of every two is a square. To find 68 two numbers such that if unity be 

 added to each of them or to their sum or to their difference, the resulting 

 sums are all squares. 



A. Martin 69 found three numbers the sum of any two of which is a 

 square and the sum of the resulting three squares is a square. Set x+y = p 2 , 

 etc. The condition p 2 -\-q 2 -{-r 2 = w 2 is satisfied if 



z +v 2 ), q = 2uv(s 2 -t 2 ), r = (s 2 - 2 )(% 2 -v 2 ), w = (s 2 +t 2 )(u 2 +v 2 }. 

 Several 70 solved a 2 -{-x = y 2 , a 2 -\-xfp = z 2 by use of 



y 2 pz 2 = a 2 pa 2 = (ampari) 2 p(amari) 2 , m? pn = 1. 



H. Brocard 71 discussed three numbers in geometrical progression, each 

 plus unity a square. 



P. W. Flood 72 found three numbers, the first two being squares, the 

 sum of all and the sum of any two being squares. Take 16z 2 , 9z 2 , y z Wxy. 

 It remains to satisfy Qx 2 I0xy+y 2 D, 16x 2 10xy-\-y 2 = D ; eliminate r". 



R. W. D. Christie 73 solved x+1 = a 2 , y+1 = 6 2 , x+y+l = c 2 , x-y+1 =d\ 

 Take e^g^-h 2 , f=2gk, a = g 2 +h 2 . Then 



, where r 2 = 5^ 2 +4 is solved by continued fractions. 

 Coccoz 74 noted that the sum and difference of any two of the three 

 numbers 2399057, 2288168 and 1873432 are squares, and gave a general 

 solution depending on a function of degree 20 [Rolle 18 ]. 



84 Application of the angular anal, to indeter. prob. degree 2, N. Y., 1848, p. 60. 



66 Math. Quest. Educ. Times, 14, 1871, 95-6. 

 68 Mathesis, 10, 1890, 174-5. 



67 Amer. Math. Monthly, 1, 1894, 96, 136, 169. 



88 Ibid., 280, 325. 



89 Math. Quest. Educ. Times, 61, 1894, 115-6. 

 10 Ibid., 65, 1896, 115. 



71 Nouv. Ann. Math., (3), 15, 1896, 288-290. 



72 Math. Quest. Educ. Times, 68, 1898, 53. 

 '"Ibid., 69, 1898,38. 



74 Illustration, July 20, 1901. Cf. Gdrardin, 85 Euler. 28 



