54] BERNOULLIAN NUMBERS FROM B^ TO 62 . 429 



In the above expression p is supposed to include every odd prime number 

 not exceeding 2n+l. 



It may be easily shewn that all the quantities 



&c. 



are integers. Hence the right-hand side of the above equation is an integer 

 which must be divisible by 2n + l and this supplies a test of the correctness 

 of the work. 



If F n = 2- 3 (<?," + Gl + Cl + &c. ) - 2 2 "- 1 + n 



where, as before mentioned, p = 2r + 1 is an odd prime number, the above 

 equation for I n may be written 



( - I)"- 1 (2w + 1) / = - {CV'J, + C 3 "I :i + &c.} + {CV'Z + C;>/ 4 + &c.} + F n . 



The reason why we assume 



= / + (-!)" (/-!), 

 instead of taking the simpler form 



= / + (-!)"/, 



is that with the above assumption the quantities / / 2 , / 3 , J 4 , / 5 , / 6 all vanish, 

 so that we have fewer quantities to calculate. 



The numbers C r n l r , which are required in order to find the value of 

 (2n + !)/, can be readily derived from the numbers C,"" 1 /,, which have been 

 already employed in finding the value of the similar quantity (2n l)I n _ 1 

 which immediately precedes it. For since 



rn _ (2n + l)2n _, _ n(2n+l) ,,_, 



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

 we have 



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



