Let $L$ be the first-order language with binary function symbol $+$, unary function symbol $E$. Let $T$ be the set of sentences over this language that are true in the natural numbers, with $+$ interpreted in the usual way, and $E(n$) interpreted as $2^n$. Is the set $T$ recursive? 



Is the structure $(\omega,+,2^n)$ undecidable? 
As Marty explained in [this answer](https://mathoverflow.net/questions/103896/beyond-presburger-arithmetic/103914#103914), this question is the central topic of the paper [On the expansion $\langle \mathbb{N},+,2^x\rangle$ of Presburger arithmetic](http://www.math.osu.edu/~friedman.8/pdf/0AppB072710.pdf), by Françoise Point, based on a joint proceedings paper with G. Cherlin on a result credited to Alexei L. Semenov. 


