CHAP. XXI] BINARY CUBIC FORM MADE A CUBE. 567 



are three ways of solving. Thus, for x z +2x-+4x+l = z 3 , z = x-\-l gives 

 x = l, z = x+2/3 gives x= 19/72, z = l+x gives x= -90/37, and each of 

 these primitive solutions furnishes new solutions as above. Cases when 

 the preceding methods fail are noted in 30; there is no rational solution 



z 3 for 



gives x = 0, while 2 = +2/3 gives the only primitive solution [von 

 Schaewen 150 noted the additional primitive solutions x= 1, x= 1/2]. 



L. Euler, 143 after reproducing ( 147-151) essentially Fermat's methods, 

 treated the new case in which a particular solution x = h, z = k, is known. 

 Taking x = h-\-y, we get a cubic whose constant term is a cube. Since 

 4+z 2 = 2 3 for x = 2 or aj = ll, we may apply the last method, or set 

 x = (2 + 2y)l(l-y} and get (8+% 2 )(l-?/) =w* or set x = (2+Uy)J(ly). 



L. Euler 144 proved that py z p 2 x z = z z is impossible if p is a prime. For, 

 z = pA, whence p 2 A 3:: Ppx s = y z . Then y = pB, whence p 2 B 3 = pA 3:: Px 3 . Then 

 x = pC, etc., and x, y, z are divisible by an indefinitely large power of p. 



W. L. Krafft 145 would make x 3 +ny z the cube of p z -{-nq z -}-ri 2 r z 3npqr by 

 setting 



which determines x, y y subject to the condition p 2 r+pq 2 -{-nqr 2 = Q, whence 



p = { q*+ ^q 4 

 Zr 



To make the radical rational, set <? = s 2 , s 6 4w 3 = 2 , whence take s 3 -M = 2/ 3 , 

 s 3 t = 2ng z . Then s*=f 3 +ng 3 , which is like the initial equation, but in 

 smaller numbers. 



P. Paoli 146 treated a-\-b 3 x z = y 3 by setting y = bx -\-m, solving the quadratic 

 in x and making the radical rational. Thus 12am 3m 4 is to be a square, 

 which he accomplished by trying values of m< >/4a. A like method was 

 stated to apply to a-\-~bx-\-c z x* = y z . 



D. M. Sensenig 147 treated without novelty ax z +bx 2 -\-cx-\-d = y 3 , when 

 a or d is a cube. 



A. Desboves 148 stated that if T = cZ z and F = cZ 2 , where T and F are 

 binary forms of the third and fourth degrees in X and Y, are such that 

 T = and F = are solvable in integers, one can determine a solution 

 (X, Y, Z) of one of the equations knowing a solution (x, y, z) of an equation 

 of the same degree by formulas giving X, Y, Z as cubic functions of x, y, z, 

 in case of T = cZ 3 , and, in case of F = cZ 2 , by functions of degree four in x, y 

 and of degree eight in z. 



143 Algebra, St. Petersburg, 2, 1770, Ch. 10, 147-161; French transl., Lyon, 2, 1774, pp. 177- 



195; Opera Omnia, (1), I, 406-414. 



144 Opera postuma, I, 1862, 217 (about 1775). 

 146 Ibid., 234. 



146 Opuscula analytica, Liburni, 1780, 128-130. 



147 The Analyst, Des Moines, 3, 1876, 104. 



148 Comptes Rendus Paris, 90, 1880, 1069. Cf . Desboves 159 of Ch. XXII. 



