Maurice Mignotte

Linear forms in two logarithms and Schneider's method. II

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Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation

We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.

ON THE DISTANCE BETWEEN ROOTS OF INTEGER POLYNOMIALS

We study families of integer polynomials having roots very close to each other.

POLYNOMIAL ROOT SEPARATION

We discuss the following question: How close to each other can two distinct roots of an integer polynomial be? We summarize what is presently known on this and related problems, and establish several new results on root separation of monic, integer polynomials.

On the distance between the roots of a polynomial

We prove a lower bound for the distance between two roots of a polynomial with complex coefficients. Such estimates are used to separate the roots of a polynomial.

Computing the measure of a polynomial

This paper is concerned with the study of the measure of an univariate polynomial. We present a collection of known results of this quantity and propose some algorithms to compute it. These algorithms are either new or variants of known ones; some are algebraic, others are semi-numerical. In the course of this study we show (or recall) how linearly recursive sequences can be used in elimination problems. We also propose algorithms for computing the number of complex zeros of a polynomial which lie outside of the unit circle. The paper ends with examples and estimates of the cost of the algorithms.

On integers with identical digits

A long-standing conjecture claims that the Diophantine equation has finitely many solutions, and, maybe, only those given by

Binomial Thue equations and polynomial powers

We explicitly solve a collection of binomial Thue equations with unknown degree and unknown S-unit coefficients, for a number of sets S of small cardinality. Equivalently, we characterize integers x such that the polynomial x 2 + x assumes perfect power values, modulo S-units. These results are proved through a combination of techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, and computational approaches to Thue equations of low degree. Along the way, we derive some new results on Fermattype ternary equations, combining classical cyclotomy with Frey curve techniques.

Fibonacci numbers at most one away from a perfect power

The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematics, combining the modular approach used in the proof of Fermat’s Last Theorem with Baker’s Theory. By contrast, using the knowledge of the all perfect powers in the Fibonacci and Lucas sequences, the determination of the perfect powers among the numbers Fn ± 1 is quite elementary.

Perfect powers with few binary digits and related Diophantine problems, II

We prove that if q > 5 is an integer, then every q-th power of an integer contains at least 5 nonzero digits in its binary expansion. This is a particular instance of one of a collection of rather more general results, whose proofs follow from a combination of rened lower bounds for linear forms in Archimedean and non-Archimedean logarithms