Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-24T01:18:22.280Z Has data issue: false hasContentIssue false
  • English
  • Français

Definability in terms of the successor function and the coprimeness predicate in the set of arbitrary integers

Published online by Cambridge University Press:  12 March 2014

Denis Richard
Denis Richard*
Affiliation:
Institute De Mathématiques Et Informatique, Université Claude Bernard (Lyon 1), 69622 Villeurbanne, France
*
Institut Universitaire de Technologie, Université de Clermont-Ferrand I, 63170 Aubière, France
Get access Rights & Permissions [Opens in a new window]

Abstract

Using 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 4 , December 1989 , pp. 1253 - 1287
Copyright
Copyright © Association for Symbolic Logic 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AW] Woods, A., Some problems in logic and number theory, and their connection, Ph.D. thesis, University of Manchester, Manchester, 1981.Google Scholar
[BV] Birkhoff, G. D. and Vandiver, H. S., On the integral divisors of an − bn , Annals of Mathematics, ser. 2, vol. 5 (1904), pp. 173–180.CrossRefGoogle Scholar
[DR1] Richard, D., Answer to a problem raised by J. Robinson: the arithmetic of positive or negative integers is definable from successor and divisibility, this Journal, vol. 50 (1985), pp. 927–935.Google Scholar
[DR2] Richard, D., All arithmetical sets of powers of primes are first order definable in terms of the successor function and the coprimeness predicate, Discrete Mathematics, vol. 53 (1985), pp. 221–248.CrossRefGoogle Scholar
[DR3] Richard, D., Définissabilité de l'arithmétigue par successeur, coprimalité et une restriction de l'addition ou de la multiplication. Comptes Rendus des Séances de l'Académie des Sciences, Série I: Mathématique, vol. 305 (1987), pp. 665–668.Google Scholar
[HS] Shapiro, H., Introduction to the theory of numbers, Wiley/Interscience, New York, 1982.Google Scholar
[JR] Robinson, J., Definability and decision problems in arithmetic, this Journal, vol. 14 (1949), pp. 98–114.Google Scholar
[PE] Erdös, P., How many pairs of products of consecutive integers have the same prime factors, American Mathematical Monthly, vol. 87 (1980), pp. 391–392.CrossRefGoogle Scholar
[RC] Carmichael, R.C., On the numerical factors of the arithmetic forms α n ± β n , Annals of Mathematics, ser. 2, vol. 15 (1913/1914), pp. 30–69.CrossRefGoogle Scholar
[RG] Guy, R., Unsolved problems in number theory. Problem Books in Mathematics, vol. 1, Springer-Verlag, Berlin, 1981.CrossRefGoogle Scholar