Do there exist known examples of predicates $P$ (possibly functional) such that

1) $P$ admits a first-order definition in the language ${\Bbb N}(+,\times,0,1)$;

2) $P$ admits no definition that does not involve both $+$ and $\times$;

3) the theory of ${\Bbb N}(+,P,0,1)$ is decidable?

(Or

3') the theory of ${\Bbb N}(\times,P,0,1)$ is decidable?

Or even better, both.)

Last I heard (a long time ago) no one can prove the undecidability of ${\Bbb N}(+,{\rm Prime}(),0,1)$, but Alan Woods showed that standard conjectures imply the definability of multiplication in this language.