648 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



= p(d+fy; thus 



2(2mp z - hx)b*+2x(2mp* - hx)b+(mp*x* - hx z - 2c*pm - 2cra 2 p) = 0. 



Equate to zero the coefficient of 6 2 . Then that of b is zero, and we obtain 

 m and h rationally in terms of p, c, x. In the special cases p = cx and 

 c=x = l, the resulting identities are simple. He gave solutions of the 

 systems formed by x 4 -\-mx 2 y 2 -\-y i = a? and various other quartics. 



Ge"rardin 186 gave solutions of a 4 +ft6 4 = c 4 +/i<2 4 for 26 numerical values of 

 h, and noted the solution a = 2p 2 , c = 2p; b, d = p^l; h = 2p z (p z l). 



SUM OF THREE BIQUADRATES NEVER A BIQUADRATE. 



L. Euler 165 - 167 stated that this theorem was hardly to be doubted, though 

 not yet proved. Again he 168 stated " It has seemed to many Geometers 

 that this theorem (x n +y n ^z n , n>2) may be generalized. Just as there do 

 not exist two cubes whose sum or difference is a cube, it is certain that it is 

 impossible to exhibit three biquadrates whose sum is a biquadrate, but that 

 at least four biquadrates are needed if their sum is to be a biquadrate, 

 although no one has been able up to the present to assign four such bi- 

 quadrates. In the same manner it would seem to be impossible to 

 exhibit four fifth powers whose sum is a fifth power, and similarly for higher 

 powers." 



Euler 187 noted that abc(a-\-b-\-c} = l has the rational solutions 4, 1/3, 

 1/6, and abcd(a+b+c+d) = l the solutions 4/3, 3/2, -1/3, -3/2. Hence 

 we cannot infer the impossibility of p 4 +g 4 +r 4 = s 4 by setting a = p 3 /qrs, 

 b = q z lprs, c = r 3 fpqs; nor that of p 5 +5 5 +r 5 +s 5 = i 5 by setting a = p 4 fqrst, -, 

 d=s*/pqrt. 



A. Desboves 188 expressed doubt as to the theorem and proved the im- 

 possibility of p 4 6p 2 g 2 7q 4 = D in connection with a study of 



which has the solutions 



X = z 2 Ti/ 2 , 7=x 2 2/ 2 , Z = 2xy, T=x 4 -y 4 . 



L. Aubry 189 proved that the fourth power of an integer ^1040 is not 

 a sum of three biquadrates. 



A. S. Werebrusow 1890 showed that no solution can be found by making 

 each term a biquadrate in Euler's identity 



SUM OF FOUR OR MORE BIQUADRATES A BIQUADRATE. 



L. Euler 190 remarked that it seemed possible to assign four biquadrates 

 whose sum is a biquadrate, but that he had found no example, whereas he 



189 Sphinx-Oedipe, 8, 1913, 13. 



187 Opera postuma, 1, 1862, 235-7 (about 1769). Cf. Euler. 249 



188 Nouv. Corresp. Math., 6, 1880, 32. Cf . Sphinx-Oedipe, 8, 1913, 27. 



189 Sphinx-Oedipe, 7, 1912, 45-6. Stated, l'interm<d. des math., 19, 1912, 48. 

 1890 L'intermSdiaire des math. 21, 1914, 161. 



190 Corresp. Math. Phys. (ed., Fuss), 1, 1843, 618 (623), Aug. 4, 1753. See preceding topic. 



