CHAPTER IX. 



SUM OF n SQUARES. 



REPRESENTATION AS A SUM OP FIVE OR MORE SQUARES. 



C. G. J. Jacob! 1 remarked that a comparison of the sixth and eighth 

 powers of two series for (2K/7r) 1/2 would yield arithmetical theorems (for 

 that from the fourth powers see Jacobi 24 - 25 of Ch. VIII). 



G. Eisenstein 2 stated that he had obtained purely arithmetical proofs 

 of these theorems of Jacobi on the representation* of numbers as the sum of 

 six or eight squares and stated the generalizations: 



The number of representations of 4r + 1 as a sum of six squares is 12s 

 and that of 4r + 3 is 20s, where s = ^>(d\ d 2 3 ), di ranging over the 

 divisors of the form 4fc + 1 of the given number, dz over the divisors 4& -f 3. 



The number of representations of an odd number as a sum of eight 

 squares equals 16 times the sum of the cubes of its divisors. 



He stated that there is no analogue for 4r + 1 of the theorem that the 

 number of representations of 4r + 3 as a sum of ten squares is 12S(d* d\}. 



Eisenstein 3 stated that, if m is an odd number > 1 having no square 

 factor, the number \f/(m) of representations of m as a sum of five squares is 

 80s, 80cr, 112s, 80<r, according as m = 1, 3, 5, 7 (mod 8), where 



the symbol being Jacobi's. For proofs see Smith 13 ' 31 and Minkowski. 28 

 Eisenstein 4 stated that the number of solutions of x\ + + x* = m is 



- 16-372 (-} 2 , M<^, ifm = 7(mod8); 



2 



8-35 m^ _2s-/* 2 M<, if m = 3 (mod 8); 



I \m/ \mj } 2 



282 (- l) (ft - 1)/2 ( - ) /i(2m - M), M odd and < m, if m = 1 (mod 4) ; 

 \mJ 



provided m has no square factor. 



V. A. Lebesgue 5 discussed the decomposition of a prime p or its double 

 into m squares, where m is a divisor > 2 of p 1. Using indices relative 

 to a primitive root of p, divide the indices of s(s -f 1) for s = 1, 2, -, 

 p 2 by m and let a , i, , a m -i be the number of the indices with the 



1 Fundamenta Nova Func. Ellip., 1829, p. 188; Werke, 1, 1881, 239. Cf. H. J. S. Smith, 



Coll. Math. Papers, 1, 1894, 306-11. Cf. Jacobi 226 of Ch. III. 



2 Jour, fur Math., 35, 1847, 135; Math. Abh., Berlin, 1847, 195. 



* One representation yields a new one if the roots of the squares are permuted or changed 

 in sign, while a composition is unaltered. 



3 Jour, fur Math., 35, 1847, 368. 



4 Jour, fur Math., 39, 1850, 180-2. 



5 Comptes Rendus Paris, 39, 1854, 593-5. 



305 



