Prove that it is impossible to find any positive integers a, b, c such that $(2a+b)(2b+a) = 2^c$

Question

Prove that it is impossible to find any positive integers a, b, c such that $(2a+b)(2b+a) = 2^c$.

This problem has been driving me crazy. Thanks for helping.

Answer 1

We have $2a+b=2^k$ and $2b+a=2^m$, where $k$ amd $m$ are non-negative integer numbers.

Thus, $a=\frac{2^{k+1}-2^m}{3}$ and $b=\frac{2^{m+1}-2^k}{3}$, which gives $k+1>m$ and $m+1>k$ and from here $k+1\geq m+1$ and $m+1\geq k+1$, which gives $k=m$, $a=\frac{2^k}{3}$, which is impossible.