All Questions
6
questions
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Congruences of partition function
I'm trying to understand Ken Ono's results showing Erdös' conjecture for the primes $\ge5$. He first shows the following: let $m\ge5$ be prime and let $k>0$. A positive proportion of the primes $\...
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1
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33
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Show the partition function $Q(n,k)$, the number of partitions of $n$ into $k$ distinct parts, is periodic mod m
Let $Q(n,k)$ be the number of partitions of $n$ into $k$ distinct parts.
I want to show that for any $m \geq 1$ there exists $t \geq 1$ such that
$$Q(n+t,i)=Q(n,i) \mod m \quad \forall n>0 \forall ...
- 103
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vote
1
answer
56
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corollary of the partition congruence
I was going though the paper of Ramanujan entitled Some properties of p(n), the number of partitions of n (Proceedings of the Cambridge Philosophical Society, XIX, 1919, 207 – 210). He states he found ...
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3
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86
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Given $n\in\mathbb{N}_{\geqslant 2}$, find the partition $(a_1,...,a_k)\in\mathbb{N}^k:\sum_{i=1}^k a_i=n$ of $n$ that maximizes $\prod_{i=1}^k a_i$
I am a solving programming question in Leetcode in which, given a number $n \in \mathbb{N}_{\geqslant 2}$, I have to find $(a_1, ..., a_k) \in \mathbb{N}^k$ such that $k \in \mathbb{N}$, $2 \leqslant ...
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174
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Compatible partitions with laws of compositions?
An exercise from Artin's Algebra:
Let S be a set with a law of composition: A partition $\Pi_1 \cup \Pi_2 \cup ...$ of S is compatible with the law of composition if for all i and j, the product ...
user198044
11
votes
2
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309
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Identity involving pentagonal numbers
Let $G_n = \tfrac{1}{2}n(3n-1)$ be the pentagonal number for all $n\in \mathbb{Z}$ and $p(n)$ be the partition function. I was trying to prove one of the Ramanujan's congruences: $$p(5n-1) = 0 \pmod 5,...
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