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



"Rotciv" 161 treated the last question for n = 3. Take the greatest integer 

 X z ^ (b a\ a 3 )/a 2 . In the first of the pair of equations, replace x z 

 by X 2 . Then if a\x\ + 03^3 = b a 2 X 2 has integral solutions X i} X 3 , the 

 required k is Xi + X 2 + X 3 . 



M. Kuschniriuk 162 proved that, if T h (m) is the number of partitions of 

 m into h parts > 0, then 



A=0 



R. D. von Sterneck 163 considered the number {n} of ways of obtaining 

 n additively from a\, a z , , using ai at most ki times, a 2 at most k 2 times, 

 etc. The number of these representations of n in which the element a,- 

 occurs at least once is 



{n - (X/b- + l)a,-} - 



AgO A^l 



where &i = &, + 1. This is used to prove that the number of representa- 

 tions of n as a sum of an odd number of distinct summands is odd if and 

 only if in the decomposition of 24n + 1 into primes either a single exponent 

 is odd and of the form 4 + 1 or no exponent is odd and there is an odd value 

 to the half sum of the exponents of those primes which are = 1, 5, 7, 11 

 (mod 24). He also found the condition that there be an odd number of 

 those representations of n by distinct summands whose number is an odd 

 multiple of 3 (or of 5 or of 7). Finally, he drew similar conclusions from a 

 general theorem due to Vahlen. 150 



A. R. Forsyth 164 expanded the reciprocal of the product 



of n pairs of factors, suppressed every term with a negative exponent for 

 any of the symbols a, b, , and in the surviving terms replaced each a, 6, 

 by unity, and proved (in accord with a conjecture communicated pri- 

 vately by MacMahon) that the sum of the resulting series is the reciprocal 

 of 



z 2 ) 2 (l - x 3 ) 2 - -(I - z") 2 (l - 



He gave a similar theorem when each pair of factors is replaced by r + 1 

 factors. 



G. B. Mathews 165 showed that the problem of multipartite partition 

 is reducible in an infinitude of ways to a problem in simple partition. For 

 example, every set of integral solutions ^ of 



ax + by + cz + dw = m, a'x -\- b'y + c'z + d'w m' 



161 L'interm6diaire des math., 3, 1896, 249-250. 



162 p r ogr., Mahr.-Triibau, 1895. Quoted from Netto, 180 128-130. 



163 Sitzungsber. Akad. Wiss. Wien (Math.), 105, Ha, 1896, 875-899. 



164 Proc. London Math. Soc., 27, 1895-6, 18-35. 



165 Ibid., 28, 1896-7, 486^90. 



