CHAP, xxii] A*+hB 4 = C 4 +hD 4 . 647 



E. Fauquembergue 178 gave the identity 



(2a 2 -15a/3-4 j S 2 ) 4 +(4a 2 +15a/3-2/3 2 ) 4 =(4a 2 +9o ;j 3+4/3 2 ) 4 +s 2 , 

 S = 4a 4 + 132a 3 0+ 17a 2 /3 2 + 132 aj S 3 +4/3 4 , 



while by Fermat's method we may make s = D in an infinitude of ways, 

 e. g., a = 8, |8 = 25. 



A. S. Werebrusow 179 gave 292 4 +193 4 = 256 4 +257 4 . 



J. E. A. Steggall 180 treated x n u n = y n v n by setting 



which determine a, b, c, X in terms of x, y, u, v. He discussed only the case 

 n = 4, whence 



4a(l+a 4 )+6(6+c)a 2 =(6+c)(6 2 +c 2 ). 



This is satisfied if b-\-c = 2a(l-\-i), and 



A particular value making the left member a square is 



8(l+a 2 ) 2 (l-18a 2 +a 4 ) 

 " (l + 14a 2 +a 4 ) 2 +64a 2 (l + a 2 ) 2 ' 



whence we derive one of Euler's tentative solutions. The smallest set of 

 integral solutions is said to be (4) . 



M. Rignaux 181 recalled [Euler 168 ] that (1) is unaltered by the substitution 



p = P+Q+R+S, q = P+Q-R-S, r = P-Q+R-S, s = P-Q-R+S. 



He obtained (p. 128, pp. 133-4) various solutions of (1). 

 A. Gerardin 1810 noted that (1) has the solution 



p = a T+ a s-2a s +a, g = 3a 2 , r = a 6 -2a 4 +a 2 +l, s 

 which is simpler than Euler's solution (5). 



E. Grigorief 182 noted that 



19 4 +5-281 4 = 417 4 +5-117 4 , 74 4 +5-101 4 = 147 4 +5-63 4 , 



the latter being erroneous. He 183 found an infinitude of solutions when 

 h = 2, the least having eleven digits (from ^ = 33, y = 13), by making special 

 assumptions leading to the condition 3w 4 2y 4 = w 2 . 



A. S. Werebrusow 184 gave 139 4 +2-34 4 = 61 4 +2-116 4 . 



A. Ge*rardin 185 treated a 4 +/i6 4 = c 4 +/id 4 by setting a c = m, d b = x, 



178 L'interm6diaire des math., 21, 1914, 17 (18-19, bibliography). 



179 Ibid., 18. 



180 Proc. Edinburgh Math. Soc., 34, 1915-6, 15-17. 



181 L'intermediaire des math., 25, 1918, 27-28. 

 la lbid., 24, 1917, 51. 



182 L'intermediaire des math., 9, 1902, 322; 10, 1903, 245. 



1 83 Ibid., 14, 1907, 184-6. 



184 Ibid., 17, 1910, 127. 



186 Sphinx-Oedipe, 6, 1911, 6-7, 11-13. Cf. Norrie. 174 



