CHAP. xvii] SYSTEM x = 2if-l, z 2 ==2z 2 --l. 487 



(1) for a = b is 



x y u v 



mb l mb(m \) m l 



G. Lemaire 16 and E. B. Escott 17 gave the solution 



c6 2 c 



*\ f*\\ 



1+6' 1+6' 



of Planude's problem. It becomes (2) for c = 6 2 --l. Escott gave two 

 particular solutions of the problem to find two parallelepipeds with equal 

 sums of sides, equal surfaces, and with volumes in a given ratio q: 



x+y+z = a+b+c, xy+yz-\-zx = ab-{-bc-\-ca, xyz = qabc. 

 See papers 438-440 of Ch. XXI. 



U. Bini 18 gave two solutions in integers of the last problem and nine 

 sets of solutions of Planude's problem, each involving a parameter. He 

 rationalized the discriminant of the quadratic with the roots x, y, satisfying 

 (1) for a = 1. 



SYSTEM x = 2y*-l, z 2 = 2z 2 -l. 



Fermat 19 stated that x = 7 is the only integral solution, excluding of 

 course the evident solution x = 1. Cf. pp. 56, 57 of Vol. I of this History. 



E. Lucas 20 wrote x = 2y 2 -w~, w=l. Then x 2 =(2y z +w 2 ) 2 -2(2ywy. 

 Multiply the latter by - 1 = I 2 - 2 I 2 . Thus x 2 = 2r 2 - s 2 , where 



r = 2y--\-w' 1 2yw, s = 2y 2 -{-w 2 4yw. 



In view of the proposed second condition, set s=d=l, whence 2 = 2r 2 s 2 

 becomes 



Also r = (?/itl) 2 +2/ 2 , since w = l. Thus r and r 2 are sums of squares of 

 consecutive integers and hence r = 5, x = 7, by papers 26-30. 



T. Pepin 21 treated 2y 2 (y 2 1) =z 2 1, obtained by eliminating x. For y 

 odd, y = a p,zl= 2a*h, z=F 1 = 8/3 2 &, whence c?p - - 1 = Shk, l=a 2 h- 4/3 2 &, 

 so that 



Thus (x 2 4:k = mh, /5 2= F/i = 4w/v, where m, n are integers maldng mn l. If 

 m = n=+l, the lower sign is excluded and the upper gives 2/i = /3 2 +o: 2 , 

 8fc = /3 2 -cr, whence cr/3 2 -l=8M becomes a 4 -27 2 = l, 7=( 2 -/3 2 )/2. The 

 case m = n = 1 leads to the same relation. This Pell equation has no 

 integral solutions except a= 1, 7 = 0. Next, let y be even, y = 2u. Then 



2 - 1) 2 , z =.f +4g 2 , dh (4^ 2 - 1) =/ 2 - 4ff 2 , 4u 2 = 4fg, 



16 L'intermediaire des math., 14, 1907, 287. 



"Ibid., 15, 1908, 11-13. 



18 Ibid., 15, 1908, 14-18. 



"Oeuvres, II, 434, 441; letters to Carcavi, Aug., 1659, Sept., 1659. Cf. C. Henry, Bull. 



Bibl. Storia Sc. Mat. e Fis., 12, 1879, 700; 17, 1884, 342, 879, letter from Carcavi to 



Huygens, Sept. 13, 1659 (extract from letter from Fermat). 



20 Nouv. Ann. Math., (2), 18, 1879, 75-6. His u, x, y are replaced by x, y, w. 



21 Atti Accad. Pont. Nuovi Lincei, 36, 1882-3, 23-33. 



