PERFECT POWERS WITH THREE DIGITS

In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and p-adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

Pillai's conjecture revisited

On the number of solutions of binomial Thue inequalities

Let $(F_{n}^{(k)})$ and $(L_{n})$ be the $k$-generalized Fibonacci and Lucas sequences, respectively. In this paper, we look at $k$-generalized Fibonacci numbers which are perfect powers of exponent larger than $1$ of Lucas numbers. That is, we deal with the Diophantine equation $$ F_{n}^{(k)}=L_{m}^{a}$$ in non-negative integers $k$, $n$, $m$, $a$, with $k\geq 3$, $m\geq 4$ and $a\geq 2$. We show that this equation has no solution under these conditions. The proof depends on lower bounds for linear forms in logarithms and some tools from Diophantine approximation.

By modifying Beukers’ proof of Apéry’s theorem that ζ ( 3 ) \zeta (3) is irrational, we derive criteria for irrationality of Euler’s constant, γ \gamma . For n > 0 n>0 , we define a double integral I n I_n and a positive integer S n S_n , and prove that with d n = LCM ⁡ ( 1 , … , n ) d_n=\operatorname {LCM}(1,\dotsc ,n) the following are equivalent: 1. The fractional part of log ⁡ S n \log S_n is given by { log ⁡ S n } = d 2 n I n \{\log S_n\}=d_{2n}I_n for some n n . 2. The formula holds for all sufficiently large n n . 3. Euler’s constant is a rational number. A corollary is that if { log ⁡ S n } ≥ 2 − n \{\log S_n\}\ge 2^{-n} infinitely often, then γ \gamma is irrational. Indeed, if the inequality holds for a given n n (we present numerical evidence for 1 ≤ n ≤ 2500 ) 1\le n\le 2500) and γ \gamma is rational, then its denominator does not divide d 2 n ( 2 n n ) d_{2n}\binom {2n}{n} . We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact log ⁡ S n \log S_n . A by-product is a rapidly converging asymptotic formula for γ \gamma , used by P. Sebah to compute γ \gamma correct to 18063 decimals.

This study investigates numbers that are powers of two and can be expressed as the sum of the squares of any two Pell numbers. We apply Baker’s theory of linear forms in logarithms of algebraic numbers, along with a variation of the Baker–Davenport reduction method, to solve the Diophantine equation [Formula: see text], where [Formula: see text], and [Formula: see text] are non-negative integers, as presented here. Furthermore, the maple[Formula: see text] codes used in the computations made throughout the paper are also shared.

On differences of perfect powers and prime powers

This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where F k is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.

Orbit Problems are a class of fundamental reachability questions that arise in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. Instances of the problem comprise a dimension $d\in \mathbb {N}$ , a square matrix $A\in \mathbb {Q}^{d\times d}$ , and a query regarding the behaviour of some sets under repeated applications of A. For instance, in the Semialgebraic Orbit Problem, we are given semialgebraic source and target sets $S,T\subseteq \mathbb {R}^{d}$ , and the query is whether there exists $n\in {\mathbb {N}}$ and x ∈ S such that Anx ∈ T. The main contribution of this paper is to introduce a unifying formalism for a vast class of orbit problems, and show that this formalism is decidable for dimension d ≤ 3. Intuitively, our formalism allows one to reason about any first-order query whose atomic propositions are a membership queries of orbit elements in semialgebraic sets. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory—Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of $\mathbb {R}^{d}$ for which membership is decidable. On the other hand, previous work has shown that in dimension d = 4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d in N, a square matrix A in Q^{d x d}, and semialgebraic source and target sets S,T subseteq R^d. The question is whether there exists x in S and n in N such that A^nx in T. The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d <= 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory - Baker's theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

On the practical solution of the Thue equation

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Recently G. Wustholz [5], [6] proved a theorem in transcendence which includes and greatly extends many classical results. In particular it generalizes Baker’s famous theorem [2] on linear forms in logarithms, and places it within the context of arbitrary commutative group varieties.