Let $p_1,p_2,...,p_n$ are some different primes $>2$ Let $k$ is solution of system $k \equiv -2^{p-1-i} \pmod{p_i}$ Than $k*2^i+1 \equiv 0 \pmod{p_i}$ And so $k*2^i+1$ is always composite

Incredibly similar to old usamo problem: https://artofproblemsolving.com/community/c6h424755p2403712