# Is it possible to get an interesting statement about even perfect numbers from the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$?

It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $$\varphi(n)$$ denotes the Euler's totient function, $$\sigma(n)=\sum_{1\leq d\mid n}d$$ the sum of divisors function and $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ is the product of distinct primes dividing $$n>1$$ with the definition $$\operatorname{rad}(1)=1$$, see the Wikipedia Radical of an integer.

One can finds the Euler's totient function and the sum of divisors function in formulations of equivalences to the Riemann hypothesis, and the so-called radical of an integer is the famous arithmetical function that appears in the formulation of the abc conjecture.

Conjecture. An integer $$n\geq 1$$ is an even perfect number if and only if $$\operatorname{rad}(n)=\frac{1}{\frac{1}{2}-2\frac{\varphi(n)}{\sigma(n)}}.\tag{1}$$

I've cited this conjecture few days ago in MSE. My intention is to know if is it possible to get some statement about the problem concerning even perfect nubmers, if there exist infinitely many of them, using the equation or well if you can argue that seems that the equation $$(1)$$ isn't useful for this purpose.

Question. Does possible to get an interesting statement about the infinitude of even perfect numbers, or a fact about them distribution, using this equation $$(1)$$ or invoking previous Conjecture (it is easy to prove that even perfect numbers $$n$$ satisfy it, but my attempt of proof for the other part of the conjecture was failed)? If you think that it isn't possible for some obstruction, please explain it. Many thanks.

You can to invoke propositions about even perfect numbers and tools or conjectures from the analytic number theory (we can search and read from the literature those statements). I hope that this is a nice exercise for this site, any case I hope comments.

• Perhaps you can use the fact that all even numbers are of the form $2^{p-1}.M_{p}$ where $M_{p}=2^{p}-1$ is a Mersenne prime. – Sylvain JULIEN Jul 10 at 10:50
• @Wojowu: I recommend against assigning gender arbitrarily to users. Your comment assumes the OP is male, for which there is no evidence; allowing our implicit gender biases to have us default to male pronouns further worsens the problem of making mathematical disciplines unwelcome for women. – Greg Martin Jul 10 at 16:42
• Many thanks Wojowu and @GregMartin , I am male and truly the best students in the degree of mathematics that I've studied were women. I hope that women get in next years the best goals in mathematics, they are our stars! – user142929 Jul 10 at 20:46