CHAP. VIII] SUM OF FOUR SQUARES. 291 



J. Liouville 44 considered an integer m all of whose prime factors are 

 = 1 (mod 4). Express 4w in all possible ways in the form (u 2 -f- v 2 } (u\ + yj), 

 where u, - , v : are odd and positive, and call two such decompositions 

 identical if and only if u = u', , Vi = v[. Denote the first factor u 2 + v z 

 by 2a. It is stated that 2a equals the number of decompositions of 16m 

 as a product of two sums of four positive odd squares. The latter number 

 exceeds Sa if m has a prime factor = 3 (mod 4). 



Liouville 45 considered the N representations of a given even integer n 

 as a sum s* + t\ + u] + v\ of four squares, where s,-, , Vi may be positive, 

 negative or zero, and two representations are distinct unless Si = s 2 , , 

 Vi = v- z . For the first squares s], we have 



zr = o ( M odd), Zs* = #, Z*: = ^#. 



i=l t=) 4 t=l O 



The second follows from nN = Zs* + + SvJ and Ss* = S$J, etc. The 

 third was verified for small values of n [proved by Stern 81 ]. By means of 

 it and n*N = Z(s* + : + v*) 2 , we get Sj=? !*! = nW/24. 

 J. G. Zehfuss 46 noted the identity 



(2o) 2 + (26) 2 + (2c) 2 + (2d} 2 = (a + b + c rf) 2 + (a + 6 - c =F d) 2 



+ (a - 6 + c =F rf) 2 + (a - 6 - c d) 2 . 



F. Pollock 47 stated that any odd number is the sum of four squares 

 the roots of two of which differ by any assigned number d from zero to the 

 maximum. For d = 0, we use a 2 + 6 2 + 2c 2 (Legendre, Theorie des nom- 

 bres, I, 186; II, 398). Next, let d = 1. Since 4n + 1 is a sum of three 

 squares, only one being odd, 



4n + 1 = (2a) 2 + (26) 2 + (2c + I) 2 , 



2n + 1 = (a + 6) 2 + (a - b} 2 + c 2 + (c + I) 2 . 



The case in which d is general is discussed by means of a special arithmetical 

 series with the general term 2n 2 + 1. 



C. Souillart 48 proved Euler's formula (1) by multiplying 



(a 2 + 6 2 + c 2 + 



by the similar determinant with p, q, r, s as first row. 



F. Pollock 49 stated that every odd number is a sum of the squares of 

 a + p + 1, a p, a + q, a q, the sum of two of which exceed the sum 

 of the remaining two by unity; also is a sum of four squares the sum of 

 whose roots is unity. 



44 Jour, de Math., (2), 2, 1857, 351-2. 



K Ibid., (2), 3, 1858, 357-360. 



48 Archiv Math. Phys., 30, 1858, 466. 



47 PhU. Trans. Roy. Soc. London, 149, 1859, 49-59. 



48 Nouv. Ann. Math., 19, 1860, 321. 



49 Phil. Trans. Roy. Soc. London, 151, 1861, 409-421. 



