CHAP, xxi] BINARY CUBIC FORM MADE A SQUARE. 569 



Given a pair of values v, s satisfying v 2 = 4s 3 +D, we can find new pairs by 

 use of the addition theorem for the elliptic function %>(u}. Only such a 

 value v is useful for which the cubic equation 153 v = Ca/f has a rational 

 root x. The simplest case D = D is treated at length and illustrated for 



L. Holzer treated 154 (x+y)(x+y 2 ) = 4Cz 3 . J. de Billy 155 (p. 41) treated 



Candido 179 of Ch. XXIII made the product of a linear and a quadratic 

 factor a cube. 



BINARY CUBIC FORM MADE A SQUARE. 



J. de Billy 155 treated many problems /= D, where/ is a cubic or quartic 

 in one or more variables with numerical coefficients. 



Fermat 156 treated 2Qx 3 +5x 2 +4Qx+lQ = z 2 . For z = 4+5z, x = l. To 

 deduce a second solution, set x = l+y. Then 



(P)f)7/\2 

 9+ 9 ) for 



From the latter, we get a third solution. 



L. Euler 157 made F=f 2 +bx+cx 2 +dx 3 = D by setting F=(f+px} 2 , where 

 2fp = b, whence x = (p 2 c)fd, or by setting F=(f+px+qx 2 ) 2 and choosing 

 p and q to make the terms in x and x 2 cancel, whence 



p = b/(2f), <?=(c-p 2 )/(2/), x=(d-2pq)/q 2 . 



But it often happens that neither of these two methods leads to a value 4= / 

 of x, as for example for/ 2 +dr 8 , and then we resort to trial. For 3+x 3 = D, 

 set x = l+y to obtain 4H ----- \-y 3 . But for 1+x 3 , x = 2+y gives 9-f-12?/ 

 -\-6y 2 -\-y* and neither of the two methods leads to a value of x other than 

 0, 2, 1; in fact, l+ 3 = CH only when x = 0, 2, 1. 



Euler 144 of Ch. XXII applied to cubics his method to make a quartic 

 a square. 



W. L. Krafft, 158 given ma?+n = b~, made mx 3 +n = z 2 by setting x = a+y, 

 z = b+3ma 2 yl(2b)=Zi or z = Zi+py 2 and in the latter case requiring that the 

 terms if shall cancel. A. J. Lexell treated the case n = k 2 by setting x ay, 

 whence (b 2 -k 2 )y* = z 2 -k 2 , and taking (bk)y 2 = zk, (b=Fk)y = z=Fk. 



L. Euler 159 noted that l+z-z 3 = D for 2 = 11/9. 



Krafft 160 made x 3 +m/ a square for relatively prime integers x, y, by 

 setting 



2T rA/n 5 ) 2 (7 = 0, 1, 2; o: 3 = l). 



163 Treated by Haentzschel, Sitzungsber. Berlin Math. Gesell., 10, 1910, 20. 



164 Monatshefte Math. Phys., 26, 1915, 289. 



165 Diophanti Redivivi, Lvgdvni, 1670, Pars Posterior. 



156 J. de Billy's Inventum novum . . . , Oeuvres de Fermat, III, 385. 



167 Algebra, St. Petersburg, 2, 1770, Ch. 8, 112-127; French transl., Lyon, 2, 1774, pp. 135- 

 152; Opera Omnia, (1), 1, 1911, 388-396. Reproduced, Sphmx-Oedipe, 1908-9, 49-57. 

 158 Euler's Opera postuma, 1, 1862, 211-2 (about 1770). 

 Ibid., 217. 

 160 Ibid., 232-4. 



