Mathematics. — “On a formula of Sytvestir’. By Prof. W. 
KAPTEYN. 
(Communicated at the meeting of November 29, 1919). 
In his paper “On the partition of numbers” Quart. Journ. of 
Math. I (1857) p. 141—152, Sy.vesrer has given a general formula 
for the number of solutions in integers (zero included) of the equation 
Oo ba, SE SEES Eo doa (Rr Don ou cee oan (1) 
where nm and a are given integers. 
Applying this formula, which is given without proof, to a partic- 
ular example, I found a fractional number. Of course this result is 
absurd. I therefore tried to construct a proof and found, as will be 
shown hereafter, that SYLveEsSTER’s formula wants a slight correction. 
If the fraction 
1 1 
D(z) (lea) (l—2m) . . (1 @) 
is developed in ascending powers of 2, it is evident that the coef- 
ficient of 2 gives exactly the number of solutions in integers of the 
equation (1). We therefore proceed to reduce this fraction to its 
partial fractions and to develop every one of these in ascending 
powers of z. The denominator being a compound quantity, the first 
thing wanted is to determine its different factors. 
Let 1—a" = 0 denote the equation containing all the prime roots 
ver) : 
of the equation 1—r" = 0, then we have 
k 
1—a" —— 7 ladi, . ° . . e e » (3) 
i=1 ————_ 
where d,,d,,...d.(d,=1,d,—=m) represent the different divisors 
of m. 
To prove this theorem let m=p%q?.. ¥, p,q,...t being prime 
numbers, then the divisors of m are the several terms of the contin- 
2 Bp ) 
(143r)(14 £0). (04 209), 
1 1 1 
One of these being 
ued product 
EE gan. 
we know that the number of prime roots of the corresponding 
equation 1—«d = 0 is 
