Definitions:

Here I present a novel conjecture using basic mathematical tools like the sum of the divisors of an integer $n$ called $\sigma(n)$, the sum of the squares of the positive divisors of n called $\sigma_2(n)$ I also use the prime-counting function which is the function counting the number of prime numbers less than or equal to some real number n. The prime-counting function is called $\pi(n)$

Conjecture:

We introduce the following expression called $A$:

$$A = \sigma_2(\pi(n) − \sigma(n + 2))$$

We focus on numbers ends with 2. I calculate $A − 1$ and so the new number ends with 1. Then I calculate the square root of this number ends with 1. When the number is an integer, it is always prime.

Example:

Let $n = 100547$, we have $A = \sigma_2(9639 − \sigma(100549)) = 8264809922$ We have $A − 1 = 8264809921$ and we calculate the square root of 8264809921 and we have $\sqrt{A−1} = \sqrt{8264809921} = 90911$ and 90911 is prime.

Questions

1. Is this conjecture interesting?
2. Is it possible to prove this conjecture or to find counter-example?

Generalization of the conjecture

If $\sqrt{A-1}$ is an integer so $\sqrt{A-1}$ is always prime.