282 Major MacMahon, Note on the parity of the number 

 These relations may be written 



l4n - 2'- a„_(8«2±^) 



i4„+i - S« a„_(8,2±35) 



^4n+2- -'«„_!_ (8«,±7^) 



*'4w+3 



s S«a 



'w-(8s2±5s) 



In fact the four relations are connected with the four elliptic 

 products ^ 



00 



n» (1 — ^ifi'w-S) (1 _ ^16W-7^ (^l _ ql6m\ 



1 



(1 _ qi6m-u^ (^i _ qi6m-5^ (1 _ ^i6m)^ 

 q H'" (1 — q^em-15^ ^i _ „i6m-i\ Q _ „i6m\ 



IT'" (1 - gi6m-i3j ^]^ _ ^i6m-3j Q _ qi6m\ 



The formulae which have been obtained soon involve high numbers 

 and are therefore suitable for the calculation of parity of p„ when 

 ni IS large. 



As an example of the use of the formulae I append the calcula- 

 tion of a.ooo, the parity of p,,,„ making use of the enumeration of 

 the partitions of n, as far as n = 200, calculated by me in connection 

 with the valuable paper by G. H. Hardy and S. Ramanujan*. 



We use the first of the four relations but we first of all require 

 the panties of p,,„ p,,„ p^^^, p^^^ ^^^^. 



-brom the third i-elation 



^250 = ftei + «60 + ^46 + «43 + ^15 + ^10 



= 1+1+0 + 1 + + 0^1. 

 From the fourth relation 



0^243 = «60 + tts? + «47 + «38 + ^18 + ^g 



= l+0+0+l-f 1+1 

 = 0. 



From the second relation 



«'241 = ^60 + «55 + ^49 + ^34 + ^22 



=1+0+1+0+0 

 = 0. 



* Proc. Lond. Math. Soc. Vol. xv et seq. 



