204 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v 



Set PQ + 1 = n(P + Q). We may discard the common factor P + Q 

 of 7 and 5, thus altering a and (3 in the same ratio, and set 7 = 4, 

 5 = p -f- Q 4 n . The first expression (2) is the square of 



(P - Q)(P + Q) + 2P(P - Q) - 4 = (P + Q)(3P - Q - 4n), 

 which is to be divided by P + Q. Hence 



2 = 3P - Q - 4n, { = 3Q - P - 4n. 



o; p 



From the above expressions for P, Q, we readily get n = 5/4. Set 

 C = 16<* 2 /3 2 , D = (9a 2 + /3 2 )(a 2 + 2 ), P = 2(9a 4 - /3 4 ). 



Then 7 = 4, 5 = D/(4a 2 /3 2 ). Suppressing the common denominator 4a 2 /3 2 



in a, 6 c, /, h #, we get 



a = a(Z> - F), 6 + c = a(C + -D), 6 - c = 0(C - D), 

 f=p(D + F), h + g = 3a(C - D), h - g = I3(C + D}. 



Euler 72 noted that 2a 2 + 26 2 - c 2 is a square if 



a = (m + n)p (m ri)q, b = (m ri)p + (m + n)q, c = 2mp 2nq. 



It suffices to make the product of the remaining two medians a square. 

 We obtain a homogeneous quartic in p, q. A special set of values making 

 it a square is found to be 



p = (m 2 + O(9ra 2 - n 2 ), q = 2wn(9m 2 + w 2 ). 

 Euler deduced his 68 two solutions and three others : 



207, 328, 145; 881, 640, 569; 463, 142, 529. 



To make a = 2x* + 2y* - z z , p = 2x z + 2z 2 - y\ 7 = W + 2z 2 - z 2 

 squares, " Atticus " 73 took x = 5n 4m, y = 2m, 2 = 2m + n. Then 

 a = (7 W 6m) 2 , and 7 = 48mn 23n z = p z determines m. Also, 



64n 2 /3 = p 4 - 50p 2 n 2 + 1649n 4 = D 



if p = n, whence m = n/2. 



J. Cunhffe 74 treated the problem subject to very special assumptions 

 and obtained for the halves of the sides 807, 466, 491. Later, he 75 gave 

 another very special treatment and obtained the sides 884, 510, 466 and 

 medians 208, 659, 683. 



N. Fuss 76 reproduced the solution in Euler's paper of 1782 with a 

 replaced by r s, /3 by r + s, 7 by p, etc. 



J. Cunliffe 77 wrote x = AC, y = BC, z = AB for the sides, and BE, 

 AF, CD for the medians. Take z x + y d. Then 



4AF 2 = 2(A 2 + AC 2 ) - BC 2 = 4z 2 + 4xy + y 2 - 4d(x + y) + 2d 2 . 



72 Posthumous paper. Comm. Arith. Coll., 2, 1849, p. 649; Opera postuma, 1, 1862, 102-3- 



73 The Gentleman's Math. Companion, London, 2, No. 9, 1806, 17. 



74 New Series of the Math. Repository (ed., Leybourn), London, 1, 1806, II, 44. 

 76 Ibid., 2, 1809, II, 31-4. 



76 M6m. Acad. Sc. St. Petersburg, 4, 1813, 247-252. 



The Gentleman's Math. Companion, London, 5, No. 27, 1824, 349-53. Extract in 1'inter- 

 nnSdiaire des math., 5, 1898, 10-11. 



