256 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VL 



quoted from A. Fleck 185 the solution 



(a 2 c + 2abd - 6 2 c) 2 + (Vd + 2abc - a 2 d) 2 = (a 2 + 6 2 ) 3 , a 2 + 6 2 = c 2 + d 2 , 



which includes the primitive solution (a 3 3a6 2 ) 2 + (3a 2 6 6 3 ) 2 = (a 2 + ft 2 ) 3 

 by Euler 6 of Ch. XX. 



* Hesse 186 gave the general solution of x- + y z = z n . 

 Several writers 187 found solutions of z 2 + y z = z 4 . 



* J. G. van der Corput 188 treated sums of two squares. 



G. H. Hardy 189 wrote r(ri) and R(ri) for the number of integral solutions 

 of /x 2 + v z = n and of /i 2 + v~ ^ n, respectively, and set R(x) = irx + P(x). 

 He proved the existence of a positive constant K such that each of 



P(x) > Kx 1 /*, P(x) < - Kx^ 



is satisfied by values of x surpassing all limit. Hence in Sierpinski's 155 result 

 P(x) = 0(x 1 / 3 ), with defined as by Landau, 179 the exponent f cannot 

 be replaced by a number < j. He gave an explicit analytic expression 

 for P(x) in terms of Bessel's functions. 



Hardy 190 proved that, for every positive e, P(x) is on the average 

 i. e. 



- f P(r) 

 xJi 



dr = 



G. Bonfantini 191 proved that, if a number n not a prime is a E], it equals 

 either a product of several factors each a E] or such a product multiplied 

 by a square which is a common factor of the given squares whose sum is n. 

 Conversely, if m is a product of several sums of two squares and if m is 

 not an even power of 2, m is a E] . 



G. Koenigs and L. Bastien 192 discussed the number of decompositions 

 of (a 2 + & 2 ) 5 asa Eh 



A. Gerardin 193 noted that & - 2/m 2 = 1 implies 



{(h - 1W + i(h - 1)V - I} 2 = 1 + {(h - 1)V + h - I} 2 . 



By means of the fact that every prime of the form 4n + 1 is a factor of 

 a number t 2 + 1, R. D. Carmichael 193a proved by Fermat's method of infinite 

 descent that such a prime is a E]. 



* A. L. Bartelds 194 gave an elementary proof of Girard's theorem. 



T. Hayashi 195 proved that y 2 + 1 =j= z* if y 4= 0. 



185 Vossische Zeitung zu Berlin, June 1, 1913. 



186 Unterrichtsblatter fur Math. u. Naturwisa., 20, 1914, 16. Haentzschel, p. 55, discussed 



Hesse's paper. 



187 Amer. Math. Monthly, 21, 1914, 199-201. 



188 Nieuw Archief voor Wiskunde, 11, 1914-5, 61. 



189 Quar. Jour. Math., 46, 1915, 263-283; Proc. London Math. Soc., (2), 15, 1916, 15-16. 

 190 Proc. London Math. Soc., (2), 15, 1916, 192-213. 



191 Suppl. al Periodico di Mat., 18, 1915, 81-6. By use of Bonfantini 142 of Ch. XIII. 



192 L'intermediaire des math., 22, 1915, 253-4; 23, 1916, 34-5. 



193 Ibid., 22, 1915, 57. 



1930 Diophantme Analysis, 1915, 39-40. 



194 Wiskundig Tijdschrift, 12, 1915-6, 159-166. 

 196 Nouv. Ann. Math., (4), 16, 1916, 150. 



