For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum\_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible to get in closed-form the parity of the arithmetical function $[A,B]$, I mean $[A,B]\text{ mod }2$. I was studying propositions about this when I wondered about the veracity of some conjectures.



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> **Question.** I would like to know if it you can to find a counterexample or well a (partial) proof for some of the following conjecture (please see bellow and comments). **Many thanks.**
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**Conjecture 2.** Let $x\geq 1$ be an integer that satisfies $[x,x]=x$, then $x$ is a prime number.


Please if these conjectures are in the literature answer the question as a reference request, or add a comment.


I relegate two conjectures to an Appendix, y deleted the old Conjecture 1 since this is false (my problem was an implementation with three integer values in my program, instead two, the
I knew the counterexample, I wrote it in my notebook, but I relate this exception with an open question; any case Conjecture 1 has a bunch of counterxamples).


**Appendix:** Adding two conjectures from the genuine version of the post. When I can I'm going to revise the old Conjecture 1.


**Conjecture 3.** Let $x,y\geq 1$ be positive integers such that $y=\sigma(x)$, $\gcd(x,y)>1$, $xy$ isn't a perfect square, $\sigma(y)\equiv0\text{ mod }2$, the integer $x$ is a triangular number and $[x,y]\equiv 1\text{ mod }2$. Then $x$ is an even perfect number.


**Conjecture 4.** If there exists an integer $N>1$ such that $[N,\sigma(N)]=3\cdot[N,N]$ holds, then $N$ is an odd perfect number.



Around the equation $\sigma\left(\square\right)=\text{prime}$: counterexamples or a proof for some of these conjectures
Please restrict to one question per post (standard policy). Here is a proof of Conjecture 2.


For any $s\in\mathbb{C}$, the function $\sigma\_s(n):=\sum\_{d\mid n} d^s$ satisfies the Hecke multiplicativity relation
$$\sigma\_s(m)\sigma\_s(n)=\sum\_{d\mid(m,n)}d^s\sigma\_s\left(\frac{mn}{d^2}\right).$$
This is straightforward to prove by restricting $m$ and $n$ to be powers of the same prime, since $\sigma\_s$ is a multiplicative function. On a deeper level, $\sigma\_s(n)$ is the $n$-th Hecke eigenvalue of a certain Eisenstein series that is a Hecke eigenform.


At any rate, setting $s=1$ and $m=n=x$, we get
$$\sigma(x)^2=\sum\_{d\mid x}d\sigma\left(\frac{x^2}{d^2}\right).$$
Assume that $x\geq 2$. Then the right-hand side is at least $\sigma(x^2)+x$, because the divisors $d=1$ and $d=x$ are present. If the left-hand side equals this, then there are no further divisors, hence $x$ is a prime number.


**P.S.** I suggest that you open a new question for Conjecture 1, another one for Conjecture 3, and a third one for Conjecture 4. This question should be closed as I proved Conjecture 2.


