CHAP. XIV] THREE SQUARES IN ARITHMETICAL PROGRESSION. 439 



A. Cunningham 30 investigated the sets of three numbers < 10000 whose 

 squares are in A. P. the ratio of the greatest to the least being as great (or as 

 small) as possible. 



W. A. Whitworth 31 noted that if three squares without a common factor 

 are in A. P., the middle one is = 1, 25 or 49 (mod 120) and each of the others 

 is =1 or 49 (mod 240). 



J. Neuberg 32 and J. Deprez 33 investigated " automedian ' triangles, 

 viz., those whose medians are proportional to the sides a, b, c. If a>b>c, 

 the condition is a 2 +c 2 = 26 2 . 



G. Bisconcini 34 noted that, if A is the common difference of three 

 squares x\ in A. P., then xlxl = A, xlxl = 2A. By the latter, Xi, 

 o* 3 = (2A : FX 2 )/(2X). Thus X = 2a 2 , A = 2aia 2 , Xi = aid2, X 3 = ai+a 2 . By 

 the first condition, xl = al+al. It is stated [incorrectly 35 ] that a 1 = r 2 s 2 , 

 a z = 2rs, whence A=4rs(r 2 s 2 ), which he called a number of Fibonacci. 



C. Botto 35 noted the incompleteness of the solution by Bisconcini. To 

 obtain all relatively prime solutions of x 2 -\-y~ = 2z 2 , note that x and y are odd, 

 and set p = (x+y}/2, q=(x y}f2. Then p 2 +q 2 = z 2 . Since p and q are 

 relatively prime, p, q = v?v 2 , 2uv, and z = u 2 +v 2 . The same substitution 

 reduces x 2 y 2 = 2z 2 to 2pq = z 2 , whence p, q = a 2 , 2b 2 and z = 2ab. 



G. Me"trod 36 noted that u 2 2v 2 = x 2 has the solutions 



u = u n (a?+2b 2 )+4v n ab, v = 2u n ab+v n (a 2 +2b 2 ), ul-2vl= -1. 

 E. Turriere 37 noted that the sides of an automedian triangle are 

 a = \(l+2t-t 2 ), 6 = X(1-M 2 ), c = \(l-2t-t 2 ). 



A. Gerardin 38 noted that for the automedian triangle with the sides 

 31, 41, 49, the sum of the sides is a square 121. J. Rose 39 noted that by 

 Turriere's formula, a+6+c = X(3 t 2 } becomes a square by choice of X. 



R. Goormaghtigh 40 restricted the last problem to relatively prime integral 

 sides, whence these are the absolute values of a 2 /3 2 2a/3, a 2 +/3 2 . The 

 perimeter is a square if a 2 +/3 2 -f 4a/3 = w 2 , whence a-\-2fi = v, v 2 = 3(3 2 +u 2 . 



See papers 15, 35, 62 of Ch. XV; 20 of Ch. XVII; 5, 8, 16 of Ch. 

 XVIII; 7, 8, 48, 49, 57, 114, 143 of Ch. XIX; 11 of Ch. XXII. On 

 x 2 +l = 2y 2 , see papers 112-129 of Ch. IV; 154, 188, 215, 234, 298 of Ch. 

 XII; 92 of Ch. XXIII. 



PAPERS WITHOUT NOVELTY ON a; 2 +3 2 =2r/ 2 . 



A. Boutin, Jour, de math, elem., (4), 4 [19], 1895, 12 [Vieta 3 ]. 

 Plakhowo, ibid., (5), 21, 1897, 95 [Frenicle 5 ]. 



30 Math. Quest. Educ. Times, 71, 1899, 56. 



31 Ibid., 72, 1900, 98. 



32 Mathesis, 9, 1889, 261-4; (3), 1, 1901, 280. 



33 Mathesis, (3), 3, 1903, 196-200, 226-30, 245-8. 



34 Periodico di Mat., 24, 1909, 157-70. 



35 Ibid., 232-4. 



38 Sphinx-Oedipe, 8, 1913, 130-1. 



37 L'enseignement math., 18, 1916, 87-8. 



38 L'intermediaire des math., 23, 1916, 173. 



39 Ibid., 24, 1917, 20-22. 



< Ibid., 88-90. 







