208 Mr Ramamijan, Some properties of p (»), 



From these 



(5) p (35m + 19) = (mod. 35) 



follows at once as a corollary. These proofs I give in § 2 and § 3. 

 I can also prove 



(4) p {Ton + 24) = (mod. 25) 



and (6) p (49n + 47) s (mod. 49), 



but only in a more recondite way, which I sketch in § 3. 



§2. Proof of {\). We have 



(11) X {(1 - cc) {l-x-){l- x^). ..Y 



= X (1 - 3« + haf -1x^^ ...){\-x-x^^ X' +...) 

 = t(- lY+^(2/ji+l)x^+^-'^'i^+'>+i''^^'' + ^\ 



the summation extending from ^ = to fi = x and from v = — oc to 

 y = 00 . Now if 



l+i/A(/x+l) + ii^(3i^ + l) = 0(mod. 5) 

 then 8 + 4/x (/x + 1) + 4z^(3!^ + 1) s (mod. 5), 



and therefore 



(12) (2ya + l)^ + 2(i; + l)^ = 0(mod. 5). 



But (2/x + 1)^ is congruent to 0, 1, or 4, and 2(v + 1)- to 0, 2, or 3. 

 Hence it follows from (12) that '2/x + l and v i-1 are both multiples 

 of 5. That is to say, the coefficient of x^^^ in (11) is a multiple of 5. 

 Again, all the coefficients in (1 — x)~^ are multiples of 5, except 

 those of 1, x^, x^°, ..., which are congruent to 1 : that is to say 



(1 — xf 1 —x" 



1 —x^ 



or -z rr = 1 (mod. 5). 



Thus all the coefficients in 



(1 ~ x') (1 - X'') (1 -x'')... 



1 



{{l-x){l-x"~){l-x') ...Y 

 (except the first) are multiples of 5. Hence the coefficient of x^'^ in 

 ^(l_^..)(l_^ao)... _ (l-a^)(l-x-) ^.^ 



{l-x)(l-x^){l-a^)...~^^ ^^ ""'^'--^ {(l-x){l-x')...Y 

 is a multiple of 5. And hence, finally, the coefficient of .r"' in 



(1 -x){l- X') (1 - x^) 

 is a multiple of 5 ; which proves (1). 



