Tìm các số nguyên dương m, n biết: 2^m + 2ⁿ = 2^m+n.

Ta có:

2^m + 2ⁿ = 2^m+n

⇔ 2^m + 2ⁿ = 2^m.2ⁿ

⇔ 2^m + 2ⁿ – 2^m.2ⁿ = 0

⇔ 2^m (2ⁿ – 1) – (2ⁿ –1) = 1

⇔ (2^m – 1)(2ⁿ – 1) = 1 = 1 . 1

Suy ra:

\(\left\{ \begin{array}{l}{2^m} - 1 = 1\\{2^n} - 1 = 1\end{array} \right.\) ⇒ \(\left\{ \begin{array}{l}m = 1\\n = 1\end{array} \right.\)

Vậy m = n = 1.