Gọi 4 số nguyên là \(a_1< a_2< a_3< a_4\) \(\Rightarrow\left\{{}\begin{matrix}a_2=a_1+d\\a_3=a_1+2d\\a_4=a_1+3d\end{matrix}\right.\) với \(d\in Z^+\)
\(a_4=a_1^2+a_2^2+a_3^2\)
\(\Leftrightarrow a_1+3d=a_1^2+\left(a_1+d\right)^2+\left(a_1+2d\right)^2\)
\(\Leftrightarrow a_1+3d=3a_1^2+6a_1d+5d^2\)
\(\Leftrightarrow3a_1^2+\left(6d-1\right)a_1+5d^2-3d=0\)
\(\Delta=\left(6d-1\right)^2-12\left(5d^2-3d\right)\ge0\)
\(\Leftrightarrow-24d^2+24d+1\ge0\Rightarrow\dfrac{6-\sqrt{42}}{12}\le d\le\dfrac{6+\sqrt{42}}{12}\)
\(\Rightarrow\left[{}\begin{matrix}d=0\left(ktm\right)\\d=1\end{matrix}\right.\) \(\Rightarrow3a_1^2+5a_1+2=0\Rightarrow\left[{}\begin{matrix}a_1=-1\\a_1=-\dfrac{2}{3}\left(ktm\right)\end{matrix}\right.\)
Vậy 4 số đó là -1; 0; 1; 2
