684 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxm 



TWO EQUAL SUMS OF WTH POWERS. 



A. Desboves 79 noted that w 5 +y 5 = s 5 +w 5 has the complex solution 



u, v = 2xy(x 2 -2y*)', s, w = 2xy 

 J. W. Nicholson 80 recalled that, if s = ai+ - 



Thus ll n = 9"+8 n +5 n -6"-3"-2 w for n = 2 or 1 [Euler 2 , Ch. XXIV]; 

 etc. 



Several writers 81 determined the signs so that 



A. de Farkas 82 proved it is impossible to find two different sets Xi and 

 y { such that for a and q are arbitrary 



N. Agronomof 83 argued the existence of integral solutions of the equation 

 xt+'-'+x^yl+'-'+y',, /^2'- 3 , g^2 p - 3 , p>4. 



But, as shown by Filippov 780 for the case p = 5, h=g = 4, the method leads 

 only to the trivial solution Xi= = x 3 , x 2 = X*, y\ = y 3 , 7/ 2 = 2/4. 



C. B. Haldeman 83a gave special rational solutions of s 3 = s 4 and s& = s n , 

 where s n denotes a sum of n fifth powers. 



On x n +v n = y n +u n , see Steggall 180 of Ch. XXII. 



MISCELLANEOUS RESULTS ON SUMS OF LIKE POWERS. 



J. Hill 84 noted that the sum of the cubes of x 2 /2, 2x 2 /3, 5z 2 /6 is a sixth 

 power a; 6 . Cf. Emerson 52 of Ch. XXI. 



L. Euler 85 stated that no sum of three biquadrates is divisible by 5 or 29, 

 which alone are exceptional. Cf. Gegenbauer 126 of Ch. XXVI. 



R. Elliott 86 noted that 1 5 H ----- h^ 5 =D if F = %(2n 2 +2n-l) = D and 

 took ri=3+l. Then 9^ = 6^+18^+9= D = (az-3) 2 determines x. The 

 anonymous proposer solved F = a 2 for n; the radical must be a rational 

 number 3c. Take a = p+q, c = p q. Then p 2 10pg+g 2 = l, whence 

 24g 2 +l = D, whose solution is known. 



G. Libri 87 expressed as a trigonometric sum the number of sets of solu- 

 tions of x"-\ -+:r"+l = (mod p), where p is a prime em+1 [Libri 147 ]. 

 Cf. pp. 224-5 of Vol. I of this History. 



79 Assoc. frang., 9, 1880, 242-4. 



80 Amer. Math. Monthly, 9, 1902, 187, 211. 



81 Math. Quest. Educat. Times, (2), 13, 1908, 110-111. 



82 L'interme'diaire des math., 20, 1913, 79-80. 

 83 T6hoku Math. Jour., 10, 1916, 211. 



830 Amer. Math. Monthly, 25, 1918, 399-402. 



"Ladies' Diary, 1737, Quest. 192; Leybourn's Math. Quest. L. D., 1, 1817, 254-5. Cf. 



Math. Quest. Educ. Times, 66, 1897, 120. 

 86 Opera postuma, I, 1862, 186 (between 1775 and 1779). 



86 Ladies' Diary, 1796, 40-1, Quest. 992; Leybourn's M. Quest. L. D., 3, 1817, 296-7. 



87 Me"m. divers savants acad. sc. de 1'Institut de France (math.), 5, 1838, 61-63. 



