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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, 63170Aubière, France
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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.

Information

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

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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