Mr Ramanujan, Some properties o/p(n) 



207 



Some properties of p (n), the number of partitions of n. By 

 S. Ramanujan, B.A., Trinity College. 



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



§ 1. A recent paper by Mr Hardy and myself* contains a table, 

 calculated by Major MacMahon, of the values oip{n), the number 

 of um-estricted partitions of n, for all values of n from 1 to 200. 

 On studying the numbers in this table I observed a number of 

 curious congruence properties, apparently satisfied by p {n). Thus 



(1) |)(4), p(9), 23(14), p(19), ... = 0(mod. 5), 



From these data I conjectured the truth of the following 

 theorem : 



If h= b'^mV and 24A, = 1 (mod. S), then 



p{\), p(\ + 8), p{X + 28),... = 0(mod.B). 



This theorem is supported by all the available evidence ; but 

 I have not yet been able to find a general proof. 



I have, however, found quite simple proofs of the theorems 

 expressed by (1) and (2), viz. 



(1) p {57n + 4) = (mod. 5) 



and (2) p (7m + 5) = (mod. 7). 



* G. H. Hardy and S. Ramanujan, 'Asymptotic formulae iu Combinatory 

 Analysis', Proc. London Math. Soc, ser. 2, vol. 17, 1918, pp. 75—115 (Table IV, 

 pp. 114—115). 



VOL. XIX. PART v. 



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