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 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 and the Modular Discriminant
We relate the parity of the partition function to the parity of the q-series coefficients of certain powers of the modular discriminant using their generating functions. This allows us to make statements about the parity of the initial values of the partition function as well as obtain a modified Euler recurrence for its parity.
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.
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2019
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042119500428