Mr Darling, On Mr Ramanujan's congruence properties of p (n) 217 



On Mr Ramanujan's congruence properties of p («). By H. B. C. 

 Darling. (Communicated by Mr G. H. Hardy.) 



[Received 3 October 1918: read 28 October 1918.] 



1. Proof that p {5ni + 4) = (mod 5). 



Let u ^(l-x){l- aJ") (1 -x')...; 



then by Jacobi's expansion 



a^' = "ST (- 1)" (2» + 1) **"^''+'^ 



n = » 



so that in d'-ur, where d denotes differentiation with respect to x, 

 the coefficients are of the form 



i {n - 1) n (n + 1) (n + 2) {2 (n + 3) - 5], 



and therefore 



d^u^= (mod 5) (1). 



Again, in d^u" the coefficients are of the form 



A^{n' + n-4<){n-2)(n-l)n(n + l)(n + 2)\2(u + 4')-7], 



and therefore 



an<3= (mod 7) (2). 



/1\ 1 2 



Now 8- ( - = /ci-u + - (duf ; 



\Uj U' u 



also du^ = Zu'du, and d-ii^ = ^u'^dhi + Qu (du)-. Hence 



a^(-) = -o\9"''' + (r7(9'*')' (3)5 



\uj ?>u^ 9«' 



and thus, by (1), we have 



^' ^ - II ^^'''^^ ^" ^ ~ 27I" ^^'''^' ^"^^^ ^^ ' 



so that 8^ f-] = (mod 5) (4). 



Again if Iju be expanded in powers of x, and the operator 3* 

 be applied to the resulting sei'ies, it is evident that the coefficients 

 of all powers of x of the forms om, 5m + 1, om + 2 and 5m + 3 will 

 be multiplied by a factor divisible by 5 ; but that the coefficients 

 of the powers of x of the form 5m + 4 will be multiplied by a factor 

 which is not divisible by 5. Hence it follows at once from (4) that 



p {oin + i) = (mod 5). 



