FWD: Votre message pour seqfan a ete rejete (fwd)

[Richard Guy]

At Subbarao's request I'm forwarding his
message.    Best to all.    R.

---------- Forwarded message ----------
Date: Thu, 4 Sep 2003 10:41:42 -0600
From: msubbara <msubbara at ualberta.ca>
To: rkg at cpsc.ucalgary.ca
Subject: FWD: Votre message pour seqfan a ete rejete

Sear Richard:
Regarding the parity of p(n) I sent yesterday the  forwarded message and it 
was returned twice. If you think that it is worth  something, please send it 
all those interested on my behalf. Thanks for your help.
With best regards,
Subbarao
m.v.subbarao at ualberta.ca

Date: Wed, 3 Sep 2003 16:37:24 -0600
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From: msubbara <msubbara at ualberta.ca>
To: bdm at cs.anu.edu.au
Cc: seqfan at ext.jussieu.fr
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Subject: FWD: RE: Parity of number of partitions of an integer
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Essentially this same question was raised
by Ramanujan in a letter to P.C.MacHahon around
the year 1920 (see page 1087, Collected Ppers of
MacMahon, edited by George Andrews published by
MIT).  With the help of Jacobi's triple product
identity, Macmahon showed (see his paper beginning
on page  1087 of above reference) that p(1000)
is odd (as he says, with five minutes work- there
were no computers those days).  Now we know that
among the first ten million values of p(n) 5002137
of them are odd.  It is conjectured (T.R.Parkin
and D.Shanks) that p(n) is equally often even and
odd.  lower bound estimates for the number of
times p(n) is enen among the first N values of
P(n) for any givan N are known((Scott Ahlgren;
and Nicolas, Rusza and Sarkozy among others).
Earlier this year a remarkable result has been
proved by Boylan and Ahlgren (AMS ABSTRACT
# 987-11-82) which says that beyond the three
eighty year old Ramanujan congruences -namely,
p(5n+4), p(7n+5) and p(11n +6) being divisible
respectively by 5,7 and 11- there are no other
simple congruences of their kind.  My 1966
conjecture that in every arithmetic progression
r(mod s) for arbitrary integral r and s, there
are infinitely many integers n for which p(n)
is odd- and a similar statement fon p(n) even,
is proved for the even case by Ken Ono (1996)
and for the odd case for all  s  up to 10^5
and also true for all  s  which are powers of 2
(Bolyan and Ono<2002).

M.V Subbarao
m.v.subbarao at ualberta.ca

>===== Original Message From John Conway <conway at Math.Princeton.EDU> =====
On Wed, 3 Sep 2003, Brendan McKay wrote:

> Is there a formula for the parity of the number of partitions
> of an integer?  It is A040051.

   This is an old question, to which nobody seems to have found even
a partial answer - a bit surprising in view of the fact that there
are simple infinite sequences of numbers for which we know the
value of  p(n)  modulo 5 or 7 or 11.

      John Conway