Odd Perfect Numbers Have At Least Nine Distinct Prime Factors

An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.Comment: 17 page

Must a primitive non-deficient number have a component not much larger than its radical?

Let $n$ be a primitive non-deficient number, with $n= p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ where the $p_i$ are distinct primes. Let $R=p_1p_2 \cdots p_k.$ We prove that there must be an $i$ such that $p_i^{a_i+1} < 4R^2$. We conjecture that there is always an $i$ such that $p_i^{a_i+1} < kR$ and prove this stronger inequality in some cases.Comment: 10 page

Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers

In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester\u27s bound stood as the best demonstrated until Gradstein improved it by one in 1925. Today, we know that the number of distinct prime divisors that an odd perfect number can have is at least eight. This was demonstrated by Chein in 1979 in his doctoral thesis. However, he published nothing of it. A complete proof consisting of almost 200 manuscript pages was given independently by Hagis. An outline of it appeared in 1980. What motivated Sylvester\u27s sudden interest in odd perfect numbers? Moreover, we also ask what prompted this mathematician who was primarily noted for his work in algebra to periodically direct his attention to famous unsolved problems in number theory? The objective of this paper is to formulate a response to these questions, as well as to substantiate the assertion that much of the modern work done on the subject of odd perfect numbers has as it roots, the series of papers produced by Sylvester in 1888

Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most $k$ of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a \BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed $k$. It is known that this implies a polynomial-time compression algorithm that turns OCT instances into equivalent instances of size at most \BigOh(4^k), a so-called kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in $k$, has turned into one of the main open questions in the study of kernelization. This work provides the first (randomized) polynomial kernelization for OCT. We introduce a novel kernelization approach based on matroid theory, where we encode all relevant information about a problem instance into a matroid with a representation of size polynomial in $k$. For OCT, the matroid is built to allow us to simulate the computation of the iterative compression step of the algorithm of Reed, Smith, and Vetta, applied (for only one round) to an approximate odd cycle transversal which it is aiming to shrink to size $k$. The process is randomized with one-sided error exponentially small in $k$, where the result can contain false positives but no false negatives, and the size guarantee is cubic in the size of the approximate solution. Combined with an \BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a reduction of the instance to size \BigOh(k^{4.5}), implying a randomized polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape