Rational points on certain quintic hypersurfaces
Rational points on curves
This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009.We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve over . The focus is on practical aspects of this problem in the case that the genus of is at least , and therefore the set of rational points is finite.
Rationale Punkte auf Fermatkurven und getwisteten Modulkurven.
Rationale Tetraeder.
Reduction of an arbitrary diophantine equation to one in 13 unknowns
Reflections on Fermat's Last Theorem.
Remarks on existentially closed fields and diophantine equations
Řešení diofantických rovnic rozkladem nad číselnými tělesy
Článek představuje zjednodušené základy teorie kvadratických zbytků a algebraické teorie čísel a jejich užití při řešení diofantických rovnic. Obsahuje i několik příkladů pro čtenáře.
Rezolution en x de la Congruence x^n+my^n==0 (mod z^n)
Siegel’s theorem and the Shafarevich conjecture
It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field and any finite set of places of , one can effectively compute the set of isomorphism classes of hyperelliptic curves over with good reduction outside . We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus would imply an effective version of Siegel’s theorem for integral points on...
Simple groups of square order and interesting sequence of primes
Solutions entières de l’équation
Soit un corps de nombres. Dans ce travail nous calculons des majorants effectifs pour la taille des solutions en entiers algébriques de des équations, , où a au moins trois racines d’ordre impair, et où et a au moins deux racines d’ordre premier à . On améliore ainsi les estimations connues ([2],[9]) pour les solutions de ces équations en entiers algébriques de .
Solutions of the Diophantine Equation from Recurrence Sequences
Consider the system , , where is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation if (or ) where and represent the sequences of Fibonacci numbers and Lucas numbers respectively....
Solving diophantine equations
Some applications of Baker's sharpened bounds to diophantine equations
Some bases of the Stickelberger ideal
Some Diophantine equations of the form .
Some diophantine equations of the form
Some diophantine equations solvable by identities
Some diophantine equations with many solutions