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.
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.
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.
Best Answer
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.