572 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



viz., y=f(p, q), x = H(p, q), H being the Hessian of/. Thus the complete 

 solution of (3), in relatively prime integers x, y, is given by a finite number 

 of pairs of quartic forms in two parameters p, q. In particular, five such 

 pairs of quartics give all solutions of z z = x*+y* in which y is odd and prime 



to x. 



R. F. Davis 177 noted that if x = p is a solution of ax z +bx+c 2 = D, two 

 further solutions are the rational roots of (apx 6) 2 = 4ac 2 (x+p). 



E. Fauquembergue 178 proved that x 2 = (y+ l)(?/ 2 +4) has no integral 

 solutions except (x, y} = (2, 0) and (10, 4), since p z q 2 -l = p*-q* implies 



p-q-l. 



A. Gerardin 179 proposed that special cubics be made squares. He and 

 L. Aubry 180 gave a partial solution for 2 3 +z 2 -hl = D. 



E. Haentzschel 1800 made use of Weierstrass' ^-function to study 



where hz and h 3 are rational or conjugate complex numbers. As an ex- 

 ample he treated Euler's 157 problem x 3 +l = D. 



For x*+x z +x+l = D see pp. 54-58 of Vol. I of this History. 



For /= D, where/ is a certain cubic, see papers 154-6 of Ch. V, 82 of 

 Ch. XV, and 163 of Ch. XXII. 



NUMBERS THE SUM OF TWO RATIONAL CUBES: 



Fermat 40 indicated a process to get an infinitude of solutions from one. 

 J. Prestet 181 employed Fermat's process to get the solution 



X = x(2if+x*), Y= -y(2x 3 +if), Z = z(x*-if). 



J. L. Lagrange 161 reduced the problem, by means of his theory of poly- 

 nomials which repeat under multiplication, to the solution of tu?-t-t?v = 

 Setting u =ft, v =fgt, and dividing by f 2 gP, we get 



Set l = l/f-llg. Then h(W-P)=4;A. Set l = kh. Then 4A/(l-& 2 ) is ft 3 , 

 so that 2A 2 (l-/b 2 ) is the cube of 2A/h. But he did not complete the dis- 

 cussion. 



L. Euler 182 proved that y = x if A = 2. 



L. Euler 183 proved the impossibility of x z +y 3 = 4z* and that the problem 

 is equivalent to the impossibility of l+2z 3 = D in rational numbers, 

 To discuss z 3 + ?/ 3 = n2 3 , set x = a+b, y = a b, z = 2v. Then a(q-+3b 2 ) = 



177 Math. Quest. Educ. Times, (2), 24, 1913, 67-8. 



178 L'intermediaire des math., 21, 1914, 81-3. 



179 Ibid., 22, 1915, 104, 128. 



180 Ibid., 23, 1916, 132-3. 



1800 Sitzungsber. Berlin Math. Gesell., 16, 1917, 85-92. 



181 Nouveaux elemens des Math., Paris, 2, 1689, 260-1. Cf. Lucas, Amer. Jour. Math., 2, 



1879, 178; Cauchy, 287 end. 



182 Algebra, 2, 1770, Art. 247; French transl., 2, 1774, pp. 355-60; Opera Omnia, (1), I, 491. 

 "'Opera postuma, 1, 1862, 243-4 (about 1782). 



