AP terms know each other

As the progression is increasing we know that the common difference $d$ exceeds $0$ , and since the terms are all integers we know that $d$ must also be an integer. So let integer $d \ge 1$ be the common difference between successive terms in the AP, and let the four different integers be $(a - d), a, (a + d)$ and $a + 2d$ for some integer $a$ .

Now the term $X$ that is the sum of the squares of the other three must be positive and the greatest in magnitude. The greatest in magnitude could either be $a - d$ or $a + 2d$ , but if it were $a - d$ then it would have to be negative, (for if it were non-negative then we would necessarily have $a + 2d \gt a - d \ge 0$ .) So we must have $X = a + 2d$ , and thus we require that

$(a - d)^{2} + a^{2} + (a + d)^{2} = a + 2d \Longrightarrow 3a^{2} + 2d^{2} = a + 2d \Longrightarrow 3a^{2} - a + 2d(d - 1) = 0$

$\Longrightarrow a = \dfrac{1 \pm \sqrt{1 - 24d(d - 1)}}{6}.$

Now for $a$ to be real we require that $24d(d - 1) \le 1$ where $d \ge 1$ , which is only the case for $d = 1$ . From this we have that $a = \dfrac{1 \pm 1}{6}$ , and since $a$ must be an integer we conclude that $a = 0$ .

The terms of the desired progression are then $-1, 0, 1, 2$ , the largest of which is $\boxed{2}.$