314 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ix 



of coefficients, we get linear equations for the c's, from which 



k\f/(ri) k\f/(n 1) k\f/(2) k\l/(l} 



k$(n 1) k\f/(n 2) k^(T) n 1 



c n 



n! 



k$(n - 2) k$(n - 3) n - 2 



1 -.. 



P. Bachmann 37 gave an exposition of the work of Smith 13 - 31 and Min- 

 kowski 28 on sums of 5 squares, and Eisenstein 2 " 4 on sums of 5, 6, 7, 8 squares. 



E. Lemoine stated and L. Ripert 38 proved that every integer equals 

 the sum of p and certain distinct squares, where p = 0, 1, 2 or 4. 



H. Delannoy 39 proved that every even square > 4 and every 4th 

 power > 1 is a sum of five squares > 0, and that a (a + 2) is a sum of 4 or 

 5 squares > 0. 



R. E. Moritz 40 considered the representation of numbers as quotients of 

 sums and differences of squares. 



O. Meissner 41 considered the representation of numbers of an algebraic 

 field as a sum of n squares. In particular, the numbers of the field defined 

 by i Vz are sums of 5 squares, 4 of which are rational. 



J. W. L. Glaisher 42 employed the sums P(m) and Q(m) of the products 

 of the roots (taken in the form 4n + 1) of the first two and three squares, 

 respectively, in each composition of 4m as a sum of 4 odd squares, and 

 proved the following theorems when m is odd. The sum of the odd roots 

 in all the representations of m as a sum of 6 squares, 3 of which are odd and 

 3 even, is 120P(w), the sign being + or according as m = 7 or 3 

 (mod 8). If a 2 + + f 2 is any partition of 2m into 6 odd squares, 

 where a, , f are taken in the form 4n + 1, and if s is the sum of the 15 

 products of a, -, f taken two at a tune, then 2s = 120 Q(m), summed 

 for all the representations of 2m by 6 odd squares. For the partitions of 8N 

 into 8 odd squares, where N is even, the corresponding sum 2s is zero. 

 The number of compositions of 8m as a sum of 8 odd squares is the sum of 

 the cubes of the divisors of m. 



K. Petr 43 proved, by use of theta functions, two hitherto unproved 

 theorems stated by Liouville on the representation of even numbers as a 

 sum of 12 or 10 squares. 



E. Jacobsthal 44 proved that every prime p = 4n + 1 is a sum 



of 6 squares, where 6 is the g.c.d. of n and p 1, and g is a primitive root 

 of p, while p ranges over a complete set of residues modulo 5. 



17 Arith. der Quad. Formen, 1898, 608-22, 652-68. 



88 Nouv. Ann. Math., (3), 17, 1898, 195-6; 19, 1900, 335-6. 



89 L'intermddiaire des math., 7, 1900, 392; 9, 1902, 237, 245. 



40 Univ. Nebraska Studies, 3, 1903, 355. Cf. Moritz 146 " of Ch. VI. 



41 Archiv Math. Phys., (3), 5, 1903, 175-6; 7, 1904, 266-8. 

 Quar. Jour. Math., 36, 1905, 349-354. 



43 Casopis, Prag, 34, 1905, 224-9. Petr. 49 



44 Anwendungen . . . quadratischen Reste, Diss. Berlin, 1906, 20. Cf. Jacobsthal, 166 Ch. VI. 



