Proyecciones Journal of Mathematics Vol. 30, N^o 3, pp. 295-302, December 2011. Universidad Católica del Norte Antofagasta - Chile

 

A Note on Büchi's Problem for p-adic numbers

 

Marianela Castillo

Universidad de Concepción, Chile

 

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ABSTRACT

We prove that for any prime p and any integer k > 2, there exist in the ring Z_p 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 Z_p. This result for k = 2 was recently obtained by J. Browkin.

AMS Classification : 11D72, 11D88.

 

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 REFERENCES

[1] J. Browkin, Buchi 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 Buchi's problem: new presentations and open problems, Zapiski Nauchn. Sem. POMI 377, pp. 111-140, (2010).

[6] T. Pheidas and X. Vidaux, Extensions of Buchi's problem: Questions of decidability for addition and n-th powers, Fundamenta Mathematicae 185, pp. 171-194, (2005).

Marianela Castillo

Universidad de Concepcion

Casilla 160-C

Concepcion

e-mail : mcastillo@udec.cl

Received : March 2011. Accepted : September 2011