430 



ON THE CALCULATION OF THE 



[54 



which may be written 



'(n-r)(2n-2r+l) 



T) n-i . 

 \ -* r y 



and a test of the correctness of the work is supplied by the divisions by 

 n r and 2n 2i-+l being performed without leaving any remainder. 



I have proved that if n be a prime number, other than 2 or 3, then 

 the numerator of the nth number of Bernoulli will be divisible by n. 



This forms another excellent test of the correctness of the work. 



I have also observed that if q be a prime factor of n, which is not 

 likewise a factor of the denominator of B n , then the numerator of B n will 

 be divisible by q. I have not succeeded, however, in obtaining a general 

 proof of this proposition, though I have no doubt of its truth. 



TABLE I. 



Formation of the quantities f n . 



2 3 7 n 330 



I + i + _L = iI 



2 + 3 23 138 



2 3 5 7 13 2730 



2 + 3 = 6 

 iii i _929 



' , i . ' . ' | ' = '5745 

 2 3 7 n 31 14322 



2 3 5 17 



n 

 10 



II 



3 

 14 

 '5 

 16 



17 



1 I ' I ' I ' I ' I ' I ' ^ 2 7843 ,g 



2 3 5 7 13 9 37 1919190 



