CHAP. VII] SUM OF THREE SQUARES. 271 



H. Schubert 84 treated x z + 2/ 2 + & u 2 , where x, y, z have no common 

 factor. They are not all odd, as seen by their residues modulo 4. Hence 

 we may assume that x and y are even, and z and u odd. Thus (z/2) 2 + (y/2) 2 

 is to be factored into \(u -f- z), %(u z}, which is done by trial. 



P. Whitworth 840 tabulated the number of ways each integer ^ 64 is a 

 sum of three squares each > 0. R. W. D. Christie noted cases of equal 

 sums of three squares. 



E. Grigorief 85 noted that [by (3)] x 2 + y 2 + 1 = z 2 is satisfied if 



2x = p z - q 2 + r 2 - s 2 , y = pq + rs, 2z = p 2 + q 2 + r 2 + s 2 , ps - rq = 1, 



when p + q + r + sis even. Escott (p. 285) listed 34 values < 500 of z. 



F. Hromadko 86 noted that n 2 + (n + I) 2 + x 2 = (x + I) 2 for 



x = n(n + 1), 



while a 2 + b 2 + x 2 = z 2 for z = x + a b, (a 6)x = 06. 



Haag 87 stated that every number not of the form (8n l)p 2 is a SO. 

 H. B. Mathieu 88 noted the identity 



(a 2 + /3 2 + 7 2 )|>V + &V + (aa + 

 = [aa/3 + 6(/3 2 + 7 2 )] 2 + 



G. Humbert 89 gave theorems on the decomposition of M + Pp into a 

 sum of three squares of such complex integers, where p = (1 + V5)/2. 



A. Hurwitz 90 noted that, if n = ^mq^q^ , where q it q 2 , are 

 prunes 4& + 3, and m is a product of powers of primes 4k + 1, 



n 2 = z 2 + ?/ 2 + s 2 

 has 



solutions. It has solutions each 4= except for n 2 = 2 2 ", 5 2 -2 2>t , since 

 n 2 = x 2 + i/ 2 has 4<r(n 2 ) solutions. 



A. S. Werebrusow 91 expressed a SI as the cube of a GO, but made errors. 



G. Bisconcini 92 gave a table of solutions of (4). 



E. Landau 93 considered the number C(x) of integers ^ x which are SI . 

 Since a positive integer is a SI if and only if it is not of the form 



/ = 4"(86 + 7), a ^ 0, 6^0, 



84 Niedere Analysis, 1, 1902, 165-6. 



840 Math. Quest. Educ. Times, (2), 1, 1902, 94-5. 



85 L'intenne'diaire des math., 10, 1903, 245. 



86 Zeitschr. Math. Naturw. Unterricht., 34, 1903, 258; 35, 1904, 305. 



87 Ibid., 35, 1904, 57. 



88 L'intermediaire des math., 11, 1904, 273. Taking a = /3 = 7 = 1 and replacing 6 by 



6 + a, we get the identity on p. 163. 



89 Comptes Rendus Paris, 142, 1906, 537. 



90 L'intermediaire des math., 14, 1907, 107. 



91 Ibid., 15, 1908, 275-6; cf. 16, 1909, 135, 256. 



92 Periodico di Mat., 22, 1907, 28-32. 



93 Archiv Math. Phys., (3), 13, 1908, 305. 



