Đặt n² – 14n – 256 = a² (a ∈ ℕ)

n² – 14n – 256 = a² \( \Leftrightarrow \) (n – 7)² – a² = 305 \( \Leftrightarrow \)(n – 7 – a)(n – 7 + a) = 305 = 1.305 = 61.5.

Vì n + a – 7 > n – a – 7 nên ta xét các trường hợp sau:

• TH1: \(\left\{ \begin{array}{l}n - a - 7 = 1\\n + a - 7 = 305\end{array} \right.\)\( \Leftrightarrow \left\{ \begin{array}{l}n = 160\\a = 152\end{array} \right.\)\(\)(TM)

• TH2: \(\left\{ \begin{array}{l}n - a - 7 = - 305\\n + a - 7 = - 1\end{array} \right.\)\( \Leftrightarrow \left\{ \begin{array}{l}n = - 146\\a = 152\end{array} \right.\) (loại)

• TH3: \(\left\{ \begin{array}{l}n - a - 7 = 5\\n + a - 7 = 61\end{array} \right.\)\( \Leftrightarrow \left\{ \begin{array}{l}n = 40\\a = 28\end{array} \right.\)\(\)      (TM)

• TH3: \(\left\{ \begin{array}{l}n - a - 7 = - 61\\n + a - 7 = - 5\end{array} \right.\)\( \Leftrightarrow \left\{ \begin{array}{l}n = - 26\\a = 28\end{array} \right.\)\(\) (loại).

Vậy n = 40; n = 160.