Abstract
AbstractUsing coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate ⊥ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are {S, ⊥}-definable over Z. Applications to definability over Z and N are stated as corollaries of the main theorem.
Publisher
Cambridge University Press (CUP)
Reference10 articles.
1. Définissabilité de l'arithmétigue par successeur, coprimalité et une restriction de l'addition ou de la multiplication;Richard;Comptes Rendus des Séances de l'Académie des Sciences, Série I: Mathématique,1987
2. All arithmetical sets of powers of primes are first-order definable in terms of the successor function and the coprimeness predicate
3. Unsolved Problems in Number Theory
4. Woods A. , Some problems in logic and number theory, and their connection, Ph.D. thesis, University of Manchester, Manchester, 1981.
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献
1. Positive Existential Definability with Unit, Addition and Coprimeness;Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation;2021-07-18
2. Defining integer-valued functions in rings of continuous definable functions over a topological field;Journal of Mathematical Logic;2020-01-30
3. Definability, decidability, complexity;Annals of Mathematics and Artificial Intelligence;1996-12