Improving the Chen and Chen result for odd perfect numbers (Q2855617)

From the text: If \(q^\alpha \) is the Euler factor of an odd perfect number \(N\), then the authors prove that its so-called index \(\sigma (N/q^\alpha ) q^\alpha \geq 3^2 \times 5 \times 7 = 315\). It follows that for any odd perfect number, the ratio of the non-Euler part to the Euler part is greater than \(3^2 \times 5 \times \)7/2.NEWLINENEWLINEThe main motivation for studying the structure of an odd perfect number is ultimately to establish that such a number cannot exist. It is known that any oddNEWLINEperfect number \(N\) must have at least 9 distinct prime factors [\textit{P. P. Nielsen}, Integers 3, A14, 9 p. (2003; Zbl 1085.11003)], be larger than 101500 [\textit{P. Ochem} and \textit{M. Rao}, Math. Comput. 81, No. 279, 1869--1877 (2012; Zbl 1263.11005)], have a squarefree core which is less than \(2N^{\frac{17}{26}}\) [\textit{F. Luca} and \textit{C. Pomerance}, New York J. Math. 16, 23--30 (2010; Zbl 1230.11008)], and every prime divisor is less than \((3N)^{1/3}\) [\textit{P. Acquaah} and \textit{S. Konyagin}, Int. J. Number Theory 8, No. 6, 1537--1540 (2012; Zbl 1272.11007)]. These results represent recent progress on what must be one of the oldest current problems in mathematics.NEWLINENEWLINEFollowing \textit{J. A. B. Dris} [Solving the odd perfect number problem: some old and new approaches, M. Sc. Thesis, De La Salle University, Philippines (2008)], in this paper we define the index \(m\) of a prime power dividing \(N\). Using a lower bound for the index one can derive an upper bound, in terms of \(N\), for the Euler factor of \(N\). Dris found the bound \(m\geq 3\); then \textit{J. A. B. Dris} and \textit{F. Luca} [A note on odd perfect numbers, preprint, \url{arxiv:1103.1437}] improved this to \(m\geq 6\). In [\textit{F.-J. Chen} and \textit{Y.-G. Chen}, Bull. Aust. Math. Soc. 86, No. 3, 510--514 (2012; Zbl 1283.11009)] a list of forms in terms of products of prime powers, which includes the results of Dris and Dris-Luca, is derived. We improve the method of Chen and Chen, obtaining an expanded list of prime power products which cannot occur as the value of an index. This enables us to conclude, in the case of the Euler factor, that \(m\geq 315\); for any other prime, if the Euler factor divides \(N\) to a power at least 2 then \(m\geq 630\), and if the Euler factor divides \(N\) to the power 1 then \(m\geq 210\).