The undecidability of the additive theory of primes (with identity) as well as the theory Th(N,+, n -> p_n), where p_n denotes the (n+1)-th prime, are open questions. As a possible approach, we extend the latter theory by adding some extra function. In this direction we show the undecidability of the existential part of the theory Th(N, +, n -> p_n, n -> r_n), where r_n is the remainder of p_n divided by n in the euclidian division.