312 HISTORY OF THE THEORY OF NUMBERS. [CHAP. IX 



T. J. Stieltjes 27 " wrote Fj(ri) for the number of decompositions of n into 

 7 squares and stated that F7(4*m)/F 7 (w) equals 



4f|.QOfc .. O QOfc+l . . 1 



3 1 , m= 1,2 (mod 4); gl , m - 3 (mod 8); 



9 



H. Minkowski 28 proved that the numbers of the form Sn + 5 are sums of 

 5 odd squares. The number of proper representations of d as a sum of 5 

 squares, not all odd, is 



summed for the integers m prime to 2d. A number d = 5 (mod 8) has 



proper representations as a sum of 5 odd squares. 



P. S. Nasimoff 29 proved that the number of decompositions of n = 2ra 

 (m odd) as a sum of 8 squares is ^-(S" 4 " 1 15)f 8 (w), where f 3 (w) is the sum 

 of the cubes of the divisors of m. He determined the number of decomposi- 

 tions of any integer into 12 squares. 



E. Cesaro 30 noted that the number of decompositions of n into v squares 

 is Ni N 2 Nz + N 4 N 5 + Ne + , where N p is the number of 

 positive integral solutions of the system of equations 



x f = p, xtfi x v y v = n. 



The numbers of decompositions of n into two and four squares increased 

 by double the number into three squares is M i M 3 + M 5 M 7 + 

 where M p is the number of positive integral solutions of xy = p, x + yrj = n. 

 H. J. S. Smith 31 proved the formula for the number of representations 

 as a sum of five squares which had been stated by him in 1867, and deduced 

 therefrom the formulas of Eisenstein. 3 The subject proposed by the Paris 

 Academy of Sciences for the Grand Prix des Sciences Math, for 1882 was 

 the theory of the representation of integers as a sum of 5 squares (with 

 citation of results of Eisenstein). Apparently no member of the commission 

 which proposed the subject of the prize knew of the earlier paper by Smith; 

 nor was the latter mentioned in the report 32 of the commission which 

 recommended that prizes of the full amount be awarded to Smith and to 



270 Comptcs Rendus Paris, 98, 1884, 663-4. 



28 M<Sm. pr6sent<5s & 1'Acad. Sc. Inst. France, (2), 29, 1884, No. 2. Gesamm. Abh., I, 1911, 



118-9, 133-4. 



29 Application of Elliptic Functions to Number Theory, Moscow, 1885. French rcsum6 in 



Annales sc. de P6cole norm. supe"r., (3), 5, 1888, 36-7. 



80 Giornalc di Mat., 23, 1885, 175. 



81 Mem. Savans Etr. Paris Ac. Sc., (2), 29, 1887, No. 1; Coll. Math. Papers, 2, 1894, 623-680; 



cf. p. 677. 

 12 Smith's Coll. Math. Papers, 1, 1894, Ixvii-lxxii. 



