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On the Additive Theory of Prime Numbers

Cégielski P., Richard D., Vsemirnov M.

Fundamenta Informaticae

2007

Vol. 81, nr 1-3

83-96

The undecidability of the additive theory of prime numbers (with identity) as well as the theory Th (N, +, n→pn), where pn denotes the (n+1)-th prime, are open questions. In a first part, we show the undecidability of Th(N, +, n→nf(n)) where f is a good approximation of the enumeration n → pn/n. In a second part, as a possible approach, we extend the former theory by adding some extra function. In this direction we show the undecidability of the existential part of the theory Th (N, +, n→pn, n→rn)), where rn is the remainder of pn divided by n in the euclidian division.