CHAP, viii] SUM OF FOUR SQUARES. 293 



F. Unferdinger 55 denoted a 2 + 6 2 + c 2 + d 2 by Za 2 and expressed 

 2a 2 -2aJ Sa^ algebraically as a SI in 48 n-1 ways, different in general. 



E. Lionnet stated and V. A. Lebesgue 56 proved that every odd number 

 is a sum of four squares of which two are consecutive. For, 4n + 1 is a SI , 

 necessarily 4q 2 + 4r 2 + (2s + I) 2 , whence 



2n + 1 = (q + r) 2 + (q - r) 2 + s 2 + (s + I) 2 . 



J. W. L. Glaisher 57 noted that, by an identity in Jacobi's Fund. Nova, 

 48a + 24a 2 + 12<x 22 + 8 3 + 2 4 + 24/3 + 12/3 2 + 4fo + 67 + 872 + 5 



equals <r(N) if N is odd, and 3<r(JV) if A 7 " is even, where a, 2 , 22 , 3 , a 4 is 

 the number of ways N is a, sum of four squares all distinct, two equal, two 

 pairs equal, three equal, four equal, respectively, while /?, /3 2 or /3 3 is the 

 number of ways N is a sum of three squares, distinct, two or three equal, 

 and 7, 7 2 , 6 are the analogous numbers for two squares and one square. 



S. R6alis 58 employed Sn + 3 = (2a - I) 2 + (26 - I) 2 + (2c - I) 2 to 

 show that 2n + 1 is the sum of the squares of 



i{/b(a-6 + c)}, i{fc(a + 6-c)}, 



%(k (- a + 6 + c)}, {& T (a + 6 + c - 2)}, 



whose sum is unity, where, ifs = a + 6 + cis even, the upper signs are 

 chosen and k = 0, while if s is odd, the lower signs are taken and k = 1. 

 More generally, every odd number N is a sum of 4 squares, the algebraic 

 sum of whose roots equals any odd number < 2 V^V. Any number 

 N = 4n + 2 is a sum a 2 + b 2 + c 2 + & 2 , where fc 2 is any chosen square < N; 

 for, according as /c is even or odd, N k 2 is of the form 4p + 2 or 4p + 1 

 and hence a S] . Also [Zehfuss 46 ], 



N = a 2 + j3 2 + 7 2 + 6 2 , 2a = a + 6 + c + &, 2/3 = - a + b - c + &, 



Hence every number Af = 4n + 2 is a sum of 4 squares the algebraic sum 

 of whose roots is any assigned one of the numbers 0, 2, 4, , 2/z, where jj? 

 is the largest square < N. Every number N = 4n + 1 (or 4n + 3) is a 

 sum of 4 squares one of which can be chosen arbitrarily among the even 

 (or odd) squares < N. 



Glaisher 59 expanded Gauss' proof of (6) and gave an arithmetical proof 

 by showing that, if A^ is odd, the number of representations of 4N as a 

 sum of 4 odd squares equals double the number of representations of Af 

 as a sum of 4 or fewer squares. 



E. Catalan 60 attributed to J. Neuberg the identity 



(a 2 + b 2 + c 2 + be + ca + a&) 2 = (a + b + c) 2 (a 2 + 6 2 + c 2 ) + (be + ca + aft) 2 . 



65 Sitzungsber. Akad. Wise. Wien (Math.), 59, II, 1869, 455-464. 

 68 Nouv. Ann. Math., (2), 11, 1872, 516-9; same by Realis. 68 



67 British Assoc. Report, 46, 1873, 11 (Trans. Sect.). 



68 Nouv. Ann. Math., (2), 12, 1873, 212-23. 



69 Phil. Mag. London, (4), 47, 1874, 443; (5), 1, 1876, 44-7. 

 60 Nouv. Corresp. Math., 1, 1874-5, 154-5. 



