CHAP. VII] SUM OF THREE SQUARES. 269 



Catalan 65 noted that (3) with 5 = does not give all solutions of 

 ^2 _ x i _j_ yz _j_ Z 2 } f or example not that with u = 27. But all primitive 

 solutions (u, x, y, z with no common factor) are said to be given by (3). 

 There are several identities giving an infinitude of (but not all) solutions 

 of (x 2 + y 2 + z 2 ) 2 = m. 



A. Desboves 66 noted that the complete solution in integers of 



X 2 + Y 2 + Z 2 = U 2 

 is given by the identity 



[2(p 2 + q 2 - s 2 )J + {2[(p - s) 2 - q 2 + p(q - 



+ [(<? - s) 2 - P 2 + 4<?(p - s)J = {3[(p - 

 Catalan 67 noted that, if x 2 + y 2 + 2 2 = 1, zx' + t/y' + zz' = 0, 

 (z' 2 + y' 2 + z*)[(yz" - zy"} 2 + (zx" - xz") 2 + (xy" - yx"} 2 } 



= (x'x"+yy'+z f z'y+{(yz"-zy")x'+(zx"-xz'W 



Catalan 68 treated u 2 = x 2 + y 2 + z 2 . Since a prime 4^ -f 1 is of the 

 form !/ 2 + z 2 , one solution is given by u = 2/j. + 1, a: = 2^u. We may set 

 ?/ + x = a 2 + /3 2 , u x = 7 2 + 5 2 and obtain a solution leading to the 

 identity (3). 



C. Hermite 69 expressed the number of decompositions of an integer into 

 3 and 5 squares in terms of the number of classes of binary quadratic forms. 



J. W. L. Glaisher 70 considered the compositions a 2 + 6 2 + c 2 , a 2 + b 2 + b 2 , 

 a 2 + a 2 + a 2 of n as a sum of three squares when n = 3 (mod 4), o, 6, c 

 being distinct odd numbers, and formed from them the respective quan- 

 tities 8aa + 86/3 + 807, 4aa + 86/3, 4aa, where a = (- l)<-i>/ 2 , . . ., 

 7 = ( l)^- 1 )/ 2 . The sum of the quantities so derived from all the com- 

 positions of n equals the expression 



<r(n) - 2ff(n - 4) + 2<r(n - 16) - 2<r(n - 36) + , 



where <r(k) is the sum of the divisors of k. This result holds also when 

 n = I (mod 4) if we use the quantities 8aa, 4aa, 4aa, aa for the respective 

 compositions a 2 + b 2 + c 2 , a 2 + 6 2 + 0, a 2 + b 2 + 6 2 , a 2 + + 0, where a 

 is odd, b and c are even, distinct and 4= 0. The number of representations of 

 n as a sum of three squares is expressed in several ways as a series involving 

 the number of representations of k as a sum of two squares. 

 E. Catalan 71 noted that 



3{(<z + 26 - I) 2 + (6 + 2a - I) 2 + (o - 6) 2 } 



= (3a - I) 2 + (36 - I) 2 + (3a + 36 - 2) 2 , 



(x 2 + y 2 + z 2 )(z' 2 + y' 2 + z' 2 ) = Z (yz" - zy"}*, 



(3) 



if x'x" + - = 1, x = x' - x"2x' 2 , - - . 



65 Bull. Acad. Roy. Belgique, (3), 9, 1885, 531. 



66 Nouv. Ann. Math., (3), 5, 1886, 232. 



67 M4m. Soc. Roy. Sc. Liege, (2), 13, 1886, 34-9 (Melanges Math. III). 

 **Ibid., (2), 15, 1888, 73-5, 211, 259 (Melanges Math. Ill, 1885, 120). 



69 Jour, fur Math., 100, 1887, 60, 63; Oeuvres, IV, 233, 237. 



70 Messenger Math., 21, 1891-2, 122-130. 



71 Assoc. fran$. av. sc., 1891, II, 195-7. 



