2181 3Ir Darling, On Mr Ramanujan's congruence properties of p {n] 



2. Proof that p {7m + 5 ) = (mod 7). 

 Differentiating (3), we have 



d' (-] = - ^, dHi' + ^^ dud-u' + A 9 (9"')' ('iiocl 7 ) 



OU SU'^ i)'U' 



= - ~dhi^ + j^^didu^y (mod 7). 

 Similarl}^ having regard to (2), 



a* (-] = ^- du^d^u^ + ^a- (du'T- (mod 7), 



d' (^) = ^. d'vPd'w + ~ d' (dtt'f (mod 7) (5), 



^0-^,(^-'y^l,^'('"'y('--^'^ ^6). 



Again a'(-^) = -3'©H-6a.Q = .-.3{..8»Q}; 

 SO that, by (5) and (6), 



36 (^] = !_' [43 (x'^d'u'd-u') + 63 {a^«3« (du'f]] (7). 



Now d{dit'')- = 2dHo'dit', 



3^ (8m3)2 = 23^ i(33«.3 + 2 (d-u'^y. 

 Thus, by (2), 



3^ (du^Y = Qd'u'^d'-u^ (mod 7) ; 



and therefore, by (7), we see that 



that is, by (2), 



^x 



38 (- j = 3 {w'd'ii'dHi'] (mod 7) ; 



= ai'dhi^dHi^ + 6w^d^ u^'d' u' (mod 7) 

 uj 



= d'u^d{£c'd-a') (mod 7) (8). 



But the coefficients in 3 (x'^a-u^) are of the form 



i (n - 1) 9i {n + 1) {n + 2) [2 (n - 3) + 7} {(// - 2) (n + 3) + 14}, 

 and are therefore divisible by 7 ; and therefore, by (8), 

 3« (-) = (mod 7). 



Hence, by considerations similar to those in the latter part of § 1, 

 we see that 



p{7m + 5) = (mod 7). 



