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A proof of Subbarao's conjecture
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Cristian-Silviu Radu
Published/Copyright:
December 3, 2011
Abstract
Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) for which p(N) is even, and infinitely many integers M ≡ r (mod t) for which p(M) is odd. In the even case the conjecture was settled by Ken Ono. In this paper we prove the odd part of the conjecture which together with Ono's result implies the full conjecture. We also prove that for every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) such that p(N) ≢ 0 (mod 3), which settles an open problem posed by Scott Ahlgren and Ken Ono.
Received: 2010-08-24
Revised: 2011-03-03
Published Online: 2011-12-03
Published in Print: 2012-11
©[2012] by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Evolution families and the Loewner equation I: the unit disc
- Minimally invasive surgery for Ricci flow singularities
- Indefinite extrinsic symmetric spaces II
- A parabolic flow toward solutions of the optimal transportation problem on domains with boundary
- A proof of Subbarao's conjecture
- Regularized theta lifts for orthogonal groups over totally real fields
- Approximation properties for free orthogonal and free unitary quantum groups
