# Combining a working hypothesis for odd perfect numbers with an inequality for logarithms

Euler's theorem for odd perfect numbers states that if there exists and odd perfect number, that is an odd positive integer $n$ satisfying $\sigma(n)=2n$, where $\sigma(m)=\sum_{d\mid m}d$ denotes the sum of divisor function, then this integer has the form $$n=p^{4\lambda+1}\cdot M^2$$ where the prime $p\equiv 1\text{ mod 4}$ and $\gcd(p,M)=1$. We assume the factorization $$n=p^{4\lambda+1}\prod_{\substack{q\text{ prime} \\ q\mid M}}q^{2e_q}$$ then it known the following

Fact. Using Euler's theorem for odd perfect numbers, the condition to be perfect is written as $$2\cdot p^{4\lambda+1}\prod_{\substack{q\text{ prime} \\ q\mid M}}q^{2e_q}=\frac{p^{4\lambda+2-1}}{p-1}\prod_{\substack{q\text{ prime} \\ q\mid M}}\frac{q^{2e_q+1}-1}{q-1},$$ since the sum of divisor function is multiplicative.

I've used the Lemma 7.9 from page 73 from Krizek, Luca and Somer, 17 Lectures on Fermat Numbers From Number Theory to Geometry Springer CMS (2001) (you can see this page as a Google Book, type here in Google Books the words: 17 lectures on fermat numbers lemma 7.9) with the specialization $y=q^{2e_q+1}-1$ and $x=p-1$ on assumption of the following

Hypothesis A. For all prime $q\mid M$ $$p<q^{2e_q+1}-1$$ holds.

I don't know if the following simple calculations were interesting. My purpose was do calculations and ask you

Question. It is possible discard from the literature previous Hypothesis A as true? What other different hypothesis could be interesting to work from my approach? I am asking with my first question, if my hypothesis to work was absurd, and with the second question what hypotheis can I work with following calculations. I hope heuristics, that you cite the literature or mathematical reasonings. Thanks in advance.

My approach was take the sum over all prime $q\mid M$ from the result of the specialization of cited Lemma 7.9, to show $$\frac{\log(p-1)}{\log p}\sum_{q\mid M}\log q^{2e_q+1}<\sum_{q\mid M}\log (q^{2e_q+1}-1),$$ and secondly take logarithms in Fact to combine showing that

Claim. On assumption of Hypothesis A one has $$\log(2p^{4\lambda+1})+\sum_{q\mid M}\log q^{2e_q}>\log\frac{p^{4\lambda+2}-1}{p-1}+\frac{\log(p-1)}{\log p}\sum_{q\mid M}\log q^{2e_q+1}-\sum_{q\mid M}\log(q-1).$$

I am not saying that previous claim be interesting.

## 1 Answer

You might want to peruse the inequalities in the paper A note on odd perfect numbers by Dris and Luca.

• You are very generous with this, and previous references, and attention. – user243301 Aug 26 '16 at 19:00
• You are very much welcome. There is also a fairly recent preprint by Brown, where questions very similar to what you ask here (specifically, regarding Hypothesis A) are considered at the very end. – Jose Arnaldo Bebita-Dris Aug 26 '16 at 21:24
• Welcome, when I can I try read in detail all information. Of course if there is a statement that previously was in the literature in some of my posts I am saying to such author: sorry me, and after I presume that such genuine ideas were her/his work. Thus very thanks much for your information, my main goal is encourage myselft to study more mathematics, and it is possible from the attention of users as you. My last thought this week is try modifications with other sequences of PROBLEMA 73 LA GACETA de la RMSE Volumen 11, número 1 (2008). – user243301 Aug 27 '16 at 7:09