Solution of Problem 1 - Online Math Gym. Teaching Math and Physics to AI and Humans.

The Problem

Rearranging the equation gives us:

$a = \frac{r^{2} - 2}{3}$ .

We will analyze three cases for $r$ : $r = 3 n$ , $r = 3 n + 1$ , and $r = 3 n + 2$ , where $n \in N$ .

(a) For $r = 3 n$ :

$r^{2} - 2 = (3 n)^{2} - 2 = 9 n^{2} - 2.$
This expression is not divisible by 3, indicating that a solution with $r = 3 n$ is not feasible.

(b) For $r = 3 n + 1$ :

$r^{2} - 2 = (3 n + 1)^{2} - 2 = 9 n^{2} + 6 n + 1 - 2 = 9 n^{2} + 6 n - 1.$
This expression is also not divisible by 3, so a solution with $r = 3 n + 1$ is not possible.

(c) For $r = 3 n + 2$ :

$r^{2} - 2 = (3 n + 2)^{2} - 2 = 9 n^{2} + 6 n + 4 - 2 = 9 n^{2} + 6 n + 2.$
This expression remains not divisible by 3, thus a solution with $r = 3 n + 2$ is also not possible.

After examining all potential cases, we conclude that there are no integer solutions.

Note: While the equation $r^{2} = 3 a - 2$ does have integer solutions for $r = 3 n + 1$ and $r = 3 n + 2$ , it does not have solutions for $r = 3 n$ .