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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 some equations over finite fields

Ioulia Baoulina (2005)

Journal de Théorie des Nombres de Bordeaux

In this paper, following L. Carlitz we consider some special equations of n variables over the finite field of q elements. We obtain explicit formulas for the number of solutions of these equations, under a certain restriction on n and q .

On systems of diophantine equations with a large number of solutions

Jerzy Browkin (2010)

Colloquium Mathematicae

We consider systems of equations of the form x i + x j = x k and x i · x j = x k , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

On the representation of numbers by quaternary and quinary cubic forms: I

C. Hooley (2016)

Acta Arithmetica

On the assumption of a Riemann hypothesis for certain Hasse-Weil L-functions, it is shewn that a quaternary cubic form f(x) with rational integral coefficients and non-vanishing discriminant represents through integral vectors x almost all integers N having the (necessary) property that the equation f(x)=N is soluble in every p-adic field ℚₚ. The corresponding proposition for quinary forms is established unconditionally.

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