146 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



L. Goldschmidt 148 gave an elementary proof of Jacobi's 30 theorem on the 

 excess (P, a, ft, ) of the number of partitions of P into an even number 

 of the a, /3, over those into an odd number of them, and showed that 



(P, 1, 2, ., m - 1) = (P, 1, 2, 3, . - ) + (P - m, 2, 3, ' ) 



+ (P -2m, 3, 4, ...)+--. 



His proof of Euler's formula (3) is essentially the same as Franklin's, 105 as 

 admitted, ibid., 39, 1894, 212. 



J. Zuchristian 149 proved, by means of Euler's recursion formula for the 

 number Uk of partitions of n into k parts, that n^ is the integer nearest to 

 (n + 3) 2 /12, while 



n 



or 



according as n is congruent to an odd or even number k modulo 12, while 

 i, = if A; * 8, r? = 1 if fc = 8. 



K. Th. Vahlen 150 wrote N(s = 2a,) for the number of partitions s = 2a z -. 

 Consider a partition s = _2e t -ai where the v elements a* are distinct. If we 

 select X of these a's, say a\, , a x , the partition may be written 



X 

 /o\ V"* ~ i v7 



(#) s = 2^ a i ~r ^Kitti, 



Consider all possible partitions (8). The excess of the number of those 

 for which X is even over the number for which X is odd is denoted by 



N(s = IX- -f 

 i 



and is proved to be zero. It suffices to prove this for the partitions (8) 

 which arise for any one s = Se^a;. From the latter we get Q) partitions 

 (8) for each X; since X has the values 0,1, 



, , 



He proved analogous formulas. Next (p. 10), from the theory of elliptic 

 functions, we have 



11(1 - z 3 "- 2 2)(l - r^-'z-'Xl - z 3n ) = Z (~ 2)*aj< 3A *-*>' 2 . 



n=l h oo 



which, if R(n) denotes the absolutely least residue of n modulo 3, may be 

 written 



ft(l - Z"^) = Z (- 2)\C (3 *'- /2 . 

 n=l A= oo 



HenceA^(s = ESU,-; (- l) fc ), for2#(ni) = /i, equals unless s = (3/i 2 - h) /2, 

 and then equals ( 1)\ Or, in words, among those partitions of s into 



148 Zeitschrift Math. Phys., 38, 1893, 121-8; Progr. d. hoheren Handelsschule, Gotha, 1892. 



149 Monatshefte Math. Phye., 4, 1893, 185-9. Cf. Glosel. 166 

 Jour, fiir Math., 112, 1893, 1-36. Cf. von Schrutka. 218 



