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On p -adic zeros of systems of diagonal forms restricted by a congruence condition

Hemar Godhino, Paulo H. A. Rodrigues (2007)

Journal de Théorie des Nombres de Bordeaux

This paper is concerned with non-trivial solvability in p -adic integers of systems of additive forms. Assuming that the congruence equation a x k + b y k + c z k d ( m o d p ) has a solution with x y z 0 ( m o d p ) we have proved that any system of R additive forms of degree k with at least 2 · 3 R - 1 · k + 1 variables, has always non-trivial p -adic solutions, provided p k . The assumption of the solubility of the above congruence equation is guaranteed, for example, if p > k 4 .

On the quasi-periodic p -adic Ruban continued fractions

Basma Ammous, Nour Ben Mahmoud, Mohamed Hbaib (2022)

Czechoslovak Mathematical Journal

We study a family of quasi periodic p -adic Ruban continued fractions in the p -adic field p and we give a criterion of a quadratic or transcendental p -adic number which based on the p -adic version of the subspace theorem due to Schlickewei.

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