A note on Büchi's problem for p-adic numbers
DOI:
https://doi.org/10.4067/S0716-09172011000300002Abstract
We prove that for any prime p and any integer k > 2,there exist in the ring Zp of p-adic integers arbitrarily long sequences whose sequence of k-th powers 1) has its k-th difference sequence equal to the constant sequence (k!); and 2) is not a sequence of consecutive k-th powers. This shows that the analogue of Buchi's problem for higher powers has a negative answer over Zp.This result for k = 2 was recently obtained by J. Browkin.
References
[1] J. Browkin, Büchi sequences in local fields and local rings, Bull. Polish Acad. Sci. Math. 58, 109-115 (2010).
[2] H. Hasse, Number Theory, Springer (1980).
[3] N. Koblitz, P-adic Numbers, p-adic Analysis, and Zeta-Functions, Springer Graduate Texts in Mathematics, (1996).
[4] J. Neukirch, Class field theory, Springer Verlag, Grundlehren der mathematischen Wissenschaften 280, A Series of Comprehensive Studies in Mathematics, (1986).
[5] H. Pasten, T. Pheidas and X. Vidaux, A survey on Büchi’s problem: new presentations and open problems, Zapiski Nauchn. Sem. POMI 377, pp. 111-140, (2010).
[6] T. Pheidas and X. Vidaux, Extensions of Büchi’s problem : Questions of decidability for addition and n-th powers, Fundamenta Mathematicae 185, pp. 171-194, (2005).
[2] H. Hasse, Number Theory, Springer (1980).
[3] N. Koblitz, P-adic Numbers, p-adic Analysis, and Zeta-Functions, Springer Graduate Texts in Mathematics, (1996).
[4] J. Neukirch, Class field theory, Springer Verlag, Grundlehren der mathematischen Wissenschaften 280, A Series of Comprehensive Studies in Mathematics, (1986).
[5] H. Pasten, T. Pheidas and X. Vidaux, A survey on Büchi’s problem: new presentations and open problems, Zapiski Nauchn. Sem. POMI 377, pp. 111-140, (2010).
[6] T. Pheidas and X. Vidaux, Extensions of Büchi’s problem : Questions of decidability for addition and n-th powers, Fundamenta Mathematicae 185, pp. 171-194, (2005).
Published
2011-12-10
How to Cite
[1]
M. Castillo, “A note on Büchi’s problem for p-adic numbers”, Proyecciones (Antofagasta, On line), vol. 30, no. 3, pp. 295-302, Dec. 2011.
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