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On the equation $\psi(-1+2(\psi(n)-n))=n$ involving the Dedekind psi function, as a...

Discussion in 'Mathematics' started by user759001, Aug 1, 2020.

  1. user759001

    user759001 Guest

    In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia Dedekind psi function, and [1]. On the other hand I add the reference that Wikipedia has the article Mersenne prime, and that I was inspired in the formula that defines the sequence A072868 from the On-Line Encyclopedia of Integer Sequences.

    The Dedekind psi function can be represented for a positive integer $m>1$ as $$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$$ with the definition $\psi(1)=1$.

    Claim. If we take $n=2^p$ with $2^p-1$ a Mersenne prime, then the equation $$\psi(2(\psi(n)-n)-1)=n\tag{1}$$ holds.

    Sketch of proof. Just direct computation using the mentioned representation for the Dedekind psi function.$\square$

    I don't know if previous equation is in the literature, one can to state a similar equation than $(1)$ involving the sum of divisors function instead of the Dedekind psi function.

    Question. I would like to know if it is possible to prove of refute that if an integer $n\geq 2$ satisfies $(1)$ then $n-1$ is a Mersenne prime. Many thanks.

    With a Pari/GP script and for small segments of integers I have not found counterexamples. I'm asking what work can be done for previous question proving the conjecture, or if you can to find a counterexample, before I'm accepting an available answer.


    [1] Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag (1976).

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