A note on Buchi´s problem for p-adic numbers

  • Marianela Castillo Universidad de Concepción.


We prove that for any prime p and any integer k > 2,there exist in the ring Zpof p-adic integers arbitrarily long sequences whose sequence of k-th powers1)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.


[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).
Cómo citar
Castillo, M. (2011). A note on Buchi´s problem for p-adic numbers. Proyecciones. Revista De Matemática, 30(3), 295-302. https://doi.org/10.4067/S0716-09172011000300002