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In this short survey paper we state the Büchi conjecture and discuss its relations with the Hilbert Tenth Problem. We give some generalizations of the conjecture, and include some numerical examples.

The circle method and pairs of quadratic forms

Henryk Iwaniec, Ritabrata Munshi (2010)

Journal de Théorie des Nombres de Bordeaux

We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.

The classification of pairs of binary quadratic forms

Jorge Morales (1991)

Acta Arithmetica

The covariety of perfect numerical semigroups with fixed Frobenius number

María Ángeles Moreno-Frías, José Carlos Rosales (2024)

Czechoslovak Mathematical Journal

Let $S$ be a numerical semigroup. We say that $h \in ℕ ∖ S$ is an isolated gap of $S$ if ${h - 1, h + 1} \subseteq S .$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by $m (S)$ the multiplicity of a numerical semigroup $S$ . A covariety is a nonempty family $𝒞$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $𝒞,$ the intersection of two elements of $𝒞$ is again an element of $𝒞$ , and $S ∖ {m (S)} \in 𝒞$ for all $S \in 𝒞$ such that $S \neq min (𝒞) .$ We prove that the set $𝒫 (F) = {S : S$ is a perfect numerical semigroup with...

The cubic congruence $x^{3} + a x^{2} + b x + c \equiv 0 (mod p)$ and binary quadratic forms $F (x, y) = a x^{2} + b x y + c y^{2}$ . II.

Tekcan, Ahmet (2010)

Acta Universitatis Apulensis. Mathematics - Informatics

This paper is concerned with the density of rational points on the graph of a non-algebraic pfaffian function.

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Tečny dvou kruhů

The (ABC) Conjecture and the radical index of integers

The abc-inequality and the generalized Fermat equation in function fields

The arithmetic of Fermat curves.

The asymptotic behaviour of the number of solutions of polynomial congruences.

The Brauer-Manin obstruction on a general diagonal quartic surface

The Büchi sequences and Hilbert's Tenth Problem

The circle method and pairs of quadratic forms

The classification of pairs of binary quadratic forms

The covariety of perfect numerical semigroups with fixed Frobenius number

The cubic congruence $x^{3} + a x^{2} + b x + c \equiv 0 (mod p)$ and binary quadratic forms $F (x, y) = a x^{2} + b x y + c y^{2}$ . II.

The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves

The density of integral points on hypersurfaces of degree at least four

The density of rational points on a pfaff curve

The density of rational points on cubic surfaces

The density of S-integral points in projective space with respect to a quadric

The difficulty of the local solubility problem for additive equations

The dimensions of integral points and holomorphic curves on the complements of hyperplanes

The Diophantine Equation ...

The Diophantine equation $a x^{2} + 2 b x y - 4 a y^{2} = \pm 1$ .

Currently displaying 1 – 20 of 149