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A note on odd partition numbers.
- Source :
- Archiv der Mathematik; Jul2024, Vol. 123 Issue 1, p39-48, 10p
- Publication Year :
- 2024
-
Abstract
- Ramanujan's partition congruences modulo ℓ ∈ { 5 , 7 , 11 } assert that p (ℓ n + δ ℓ) ≡ 0 (mod ℓ) , where 0 < δ ℓ < ℓ satisfies 24 δ ℓ ≡ 1 (mod ℓ). By proving Subbarao's conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes ℓ ≥ 5 , we give a new proof of the conclusion that there are infinitely many m for which p (ℓ m + δ ℓ) is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such m satisfies m < (ℓ 2 - 1) / 24 , representing a significant improvement to the previous bound. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0003889X
- Volume :
- 123
- Issue :
- 1
- Database :
- Complementary Index
- Journal :
- Archiv der Mathematik
- Publication Type :
- Academic Journal
- Accession number :
- 178029046
- Full Text :
- https://doi.org/10.1007/s00013-024-01999-7