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This paper deals with conditions of compatibility of a system of copulas and with bounds of general Fréchet classes. Algebraic search for the bounds is interpreted as a solution to a linear system of Diophantine equations. Classical analytical specification of the bounds is described.

Bounds of prime solutions of some diagonal equations.

Ming-Chit Liu (1982)

Journal für die reine und angewandte Mathematik

Büchi's problem in any power for finite fields

Hector Pasten (2011)

Acta Arithmetica

Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields

István Gaál, Gábor Petrányi (2014)

Czechoslovak Mathematical Journal

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $ξ$ of the polynomial $P_{t} (x) = x^{4} - t x^{3} - 6 x^{2} + t x + 1$ , assuming that $t > 0$ , $t \neq 3$ and $t^{2} + 16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal...

Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)

Yuri F. Bilu (2005)

Journal de Théorie des Nombres de Bordeaux

This is an exposition of the recent work of Bugeaud, Hanrot and Mihăilescu showing that Catalan’s conjecture can be proved without using logarithmic forms and electronic computations.

Catalanova domněnka dokázána

Yann Bugeaud, Maurice Mignotte (2005)

Pokroky matematiky, fyziky a astronomie

Catalan’s conjecture

Yuri F. Bilu (2002/2003)

Séminaire Bourbaki

The subject of the talk is the recent work of Mihăilescu, who proved that the equation $x^{p} - y^{q} = 1$ has no solutions in non-zero integers $x, y$ and odd primes $p, q$ . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebratedconjecture of Catalan (1843): the only solution to $x^{u} - y^{v} = 1$ in integers $x, y > 0$ and $u, v > 1$ is $3^{2} - 2^{3} = 1$ . Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute...

Catalan's equation has no new solution with either exponent less than 10651.

Mignotte, Maurice, Roy, Yves (1995)

Experimental Mathematics

Characterizing the sum of two cubes.

Broughan, Kevin A. (2003)

Journal of Integer Sequences [electronic only]

Chebyshev polynomials and Pell equations over finite fields

Boaz Cohen (2021)

Czechoslovak Mathematical Journal

We shall describe how to construct a fundamental solution for the Pell equation $x^{2} - m y^{2} = 1$ over finite fields of characteristic $p \neq 2$ . Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation $x^{2} - m y^{2} = n$ .

CHRONIQUE (vol. 1, Fasc. 2.)

(1948)

Colloquium Mathematicae

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