CHAP, xvil] SYSTEM x = 2y*-l, z2 = 2z2-l. 487

(1) for a = b is

x y u v

mb — 1 mb(m — l) m—l m(mb — G. Lemaire16 and E. B. Escott17 gave the solution

cb* c

of Planude's problem. It becomes (2) for c=62— 1. 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, xyjryz+zx = ab+bcjrca, xyz — qabc. See papers 438-440 of Ch. XXI.

U. Bini18 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 o = l.

SYSTEM x = 2y2— 1, z2 = 2z2— 1.

Fermat19 stated that # = 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. Lucas20 wrote x = 2y2-w~, 10= ±1. Then z2 = (2^2+w2)2~2(2^)2. Multiply the latter by - 1 = I2 - 2 • I2. Thus x2 = 2r2 - s2, where

r~2y2+w2-2yw, s = 2y2+w2-4yw.

In view of the proposed second condition, set s = =tl, whence rc2 = 2r2 — s2 becomes

Also r = (2/±l)2-|-2/2, since w==bl. Thus r and r2 are sums of squares of consecutive integers and hence r = 5, z = 7, by papers 26-30.

T. Pepin21 treated 2y2(y2 — 1) =22 — 1, obtained by eliminating x. For y odd, y = a/3, z±l = 2a2/i, 2^1 = 8/3^, whence a2/32 - 1 = Shk, ±l = a2/i-so that

Thus «2db4A;==m/i, /32::F/i = 4nA;, where m, n are integers making mn~l. If m = n=4-l, the lower sign is excluded and the upper gives 2/i = /32+a2, 8k = p- a2, whence a2/32 - 1 = 8Wfc becomes a4 - 2T2 = 1, y = (a2 - /32) /2. The case m = 7i = — 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

z2 = (2u)4+ (4u2 - 1)2, z =Jl2+4^2, ± (4^2 - 1) =/2 - 4fl2,

18 L'intermSdiaire dea math., 14, 1907, 287.

«7&id., 15, 1908, 11-13.

" 7Wd., 15, 1908, 14-18.

190euvres, 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. 31 Atti Accad. Pont. Nuovi Lincei, 36, 1882-3, 23-33.