Contribution à l'étude d'une conjecture de théorie des nombres par le codage ZBV. (Contribution to the study of a conjecture of number theory by ZBV coding)
The paper is devoted to the problem formulated by \textit{J. Robinson} in ``Definability and decision problems in arithmetic'' [J. Symb. Logic 14, 98-114 (1949; Zbl 0034.008)]: can the first order arithmetic of natural numbers be defined in terms of the successor function S and the coprimeness relation \(\perp\). There are several results of J. Robinson and D. Richard, respectively, which show that the arithmetic is definable in terms of S, \(\perp\) and one of the following relations: \(x<y\); \(x+y=z\); \(x| y\); \(x\times y=z\), \(y=m^ x\) (for any fixed \(m\geq 2)\). It turns out that the question of J. Robinson is connected with some open problems in number theory. The aim of the reviewed paper is to present some results from number theory which can be useful to develop methods relevant to mathematical logic, to give a survey of results concerning arithmetical definability and to show that there are more relations which together with S and \(\perp\) suffice to define arithmetic, namely RES(x,y), which means that p is prime and x is a quadratic residue modulo p, and POW(y,x), meaning that y is a power of x, as well as weak restrictions of addition, multiplication and division.
scientific article; zbMATH DE number 4121990
scientific article