An Impossible Sequence of Prime Powers

Question

Let $x_1,x_2,\ldots$ be a sequence of positive integers that satisfies the recurrence relation $$x_{n+1}=2x_n(x_n-1)+1$$ for all positive integers $n$. It seems impossible that every term in this sequence is a power of a prime, but I can't see a way to prove this. I have tried rewriting the terms of the sequence in terms of each other and doing several basic algebraic manipulations, but I seem to be stuck. Does anyone see a way to prove that the terms of the sequence cannot all be prime powers?

Answer 1

Not really an answer, but I found an explicit formula for $x_n$ in terms of $x_0$ and I thought that it might be relevant: $$x_n=2^{2^{n+1}-1}(x_0-1/2)^{2^{n+1}}+1/2$$ or, rewritten, $$x_n=\frac{(2x_0-1)^{2^{n+1}}+1}{2}$$ Perhaps this will help you with your proof.