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



F. L. Griffin and G. B. M. Zerr 272 made a sum of n squares a biquadrate. 

 A. Ge"rardin 273 noted that s 4 is divisible by Sz, where 



For/=l, the quotient is 4175. E. Fauquembergue noted also that 

 5 4 +3 4 -6 4 = 59(5 3 +3 3 -6 3 ), 5 4 +6 4 -7 4 = 240(5 3 +6 3 -7 3 ). 



A. Cunningham 274 expressed numbers in the form (x 6 +2/ 6 )/(^ 2 +2/ 2 ) in 

 several ways. 



A. Cunningham 275 found certain types of solutions of 



f(x, y)+f(x', </')=*& *)+(*', V), f(x, 2/)= 



and (pp. 111-2) of s(x, y)=s(x, 2), 6xy=n, y^z. He 276 gave various 

 criteria for the solvability of Nx\2yl=xl2y\, JV=1 (mod 8). 

 He discussed (p. 108) gi(?2<78=D, where g r =z*+?/*. He 277 proved the 

 existence of an infinitude of integral solutions of F(x, y] =F(x', y'} for each 

 a/k, where 



F(x, y)=ax*-\-4:ax 3 y+kx 2 y'*-\-4:axy 3 -{-ay 4 . 



If (k+Wa)f(k 6a) is a rational square, F(x, y} is a product of two factors. 

 If (pp. 94-95) either of (2a=F26+c)(12a-2c) is of the form -a 2 -/3 2 , 



(a;, y) = ax* + bx 3 y + cx*y z + bxy 3 +ay* = <f> (x' } y'} 



is usually solvable in integers. Certain numbers (pp. 39-40) can be 

 expressed simultaneously in the forms 



_x\-y\ _z\+x\ _zlyl 



iVl - - JV2- IV 3 



, 



23+2/3 



and ATl/3, ^/3, J^i/3, JVl/3, where N{ = (x'^-y'^Kx, -y\\ etc. He 278 con- 

 sidered numbers expressible in two or four of the forms (x 4 2?/ 2 ), 

 (x 2 2i/ 4 ). He 279 showed that certain binary quartic functions of four 

 pairs of variables are equal for an infinite of set of values, by use of the 

 above 275 s(, 77). 



He 280 solved Ni+N 2 = N s +Nt, where N r =(xl-y^l(x r -y r ). 



He 281 gave a method to solve x 3 yxy z = a. 



H. B. Mathieu 282 noted that each triangular number which is a square 

 yields a solution of x 3 -\-y 2 =z*. Thus, A 49 = 35 2 gives 



49 3 +1176 2 = 35 4 . 



272 Amer. Math. Monthly, 17, 1910, 147-8. 



273 Sphinx-Oedipe, 1906-7, 159-160. 



27 < Mess. Math., 39, 1909-10, 97-128; 40, 1910-11, 1-36. 

 276 Math. Quest. Educ. Times, (2), 16, 1909, 75. 



276 Ibid., (2), 17, 1910, 66-7. 



277 Ibid., (2), 19, 1911,27-28. 



278 /bid., (2), 22, 1912, 40-41, 107-9; 23, 1913, 62-6. 

 Ibid., (2), 21, 1912, 89-90, 103-4. 



280 Ibid. t (2), 26, 1914, 60. 



281 Ibid., (2), 27, 1915, 74-5. 



282 L'interme'diaire des math., 19, 1912, 129. 



