منابع مشابه
Parity of the Partition Function
Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus t whe...
متن کاملA Combinatorial Proof on Partition Function Parity
One of the most basic results on the number-theoretic properties of the partition function p(n) is that p(n) takes each value of parity infinitely often. First proved by Kolberg in 1959, this statement was strengthened by Kolberg and Subbarao in 1966 to say that both p(2n) and p(2n + 1) take each value of parity infinitely often. These results have received several other proofs, each relying to...
متن کاملParity of the Partition Function in Arithmetic Progressions, Ii
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. We prove the conjecture for every arithmetic progression whose modulus is a power of 2.
متن کاملSubbarao’s Conjecture on the Parity of the Partition Function
Let p(n) denote the ordinary partition function. In 1966, Subbarao [18] conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N (resp. M) ≡ r (mod t) for which p(N) is even (resp. odd). We prove Subbarao’s conjecture for all moduli t of the form m · 2 where m ∈ {1, 5, 7, 17}. To obtain this theorem we make use of recent results of Ono and Taguchi [14] on ...
متن کاملParity of the Partition Function in Arithmetic Progressions
Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [23]). In this paper we prove that there indeed are infinitely many integers N in every arithmetic progression for which p(N) i...
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ژورنال
عنوان ژورنال: Comptes Rendus Mathematique
سال: 2019
ISSN: 1631-073X
DOI: 10.1016/j.crma.2019.05.006