210 Mr Ramanujan, Some properties of p (n) 



of |, forming two arithmetical progi'essions with common difl-erence 

 5. It follows that 



(15) (l-oc^) (1 - x^°) (1 - X'') ...{p{4<)+p(9)a; + jj(14).7;-+ . ..} 



5 . 



Again, if in (14) we substitute cox% (o^x% (o"x\ and q}*x% where 

 w" = 1, for x^, and multiply the resulting five equations, we obtain 



\ (l-ay^){l-x^^)(l-x^^)... Y^ 1 



^^^ \(l-x){l-x'^)(l-x')... I ^-'-Ux-x'^-^- 

 From (15) and (16) we deduce 

 (17) ^(4)+_p(9)^'+_p(14)a;-+ ... 



_ {(1 - x'){l - x'^yj l-x^')...}' . 



I 



{(1 -x){l - x')(l-a^) ...Y ' 

 from which it appears directly that ^ (5m + 4) is divisible by 5. 

 The corresponding formula involving 7 is 



(18) p(5)+p(12)x + p(19)x''+ ... 



{(1-^)(1-^'^)(1-^^)...}^ 



- , ^9^ { a-x')(l-x^^)(l-af^)...Y 

 ^ {{l-x)(l-x'){l-x'')...Y ' 



Avhich shows that p (7m + 5) is divisible by 7. 

 From (16) it follows that 

 p (4) X + p (9) x'- + p (14!) x^ + ... 



5{{l-x^){l-x^'>)(l-x'')...Y 



X {l-x'){l-x'">){l-x'')., 



{l-x)(l-x'')(l-af)... {(l-x){l-x^){l-x')...Y' 



As the coefficient of «^'* on the right-hand side is ^ multiple of 5, it 

 follows that p {25m + 24) is divisible by 25. 

 Similarly 

 p(5)x + p (1 2)x'-+p(19)x^+ ... 



7{(1-«0(1-^")(1-^'')---}' 



(l-x')(l-x^')... 



= x(l-3x+5x^- 7a'« + . . .) 



Ki-^)(i-^^)-r 



^^ja-^)(i-^^) 



{(1-^)(1-^-) 



18 ' 



"J 



from which it follows that p {4<9m + 47) is divisible by 49. 



[Another proof of (1) and (2) has been found by Mr H. B. C. Darhng, to whom 

 my conjecture had been communicated by Major MacMahon. This proof will also 

 be published in these Proceedings. I have since found proofs of (3), (7), and (8).] 



