Diophantine equations with products of consecutive terms in Lucas sequences

In this paper, we show that if (u_n)_n?1 is a Lucas sequence, then the Diophantine equation in integers n?1, k?1, m?2 and y with |y|>1 has only finitely many solutions. We also determine all such solutions when (u_n)_n?1 is the sequence of Fibonacci numbers and when u_n=(xⁿ-1)/(x-1) for all n?1 with some integer x>1.     

Diophantine equations with products of consecutive values of a quadratic polynomial

Let a, b, c, d be given nonnegative integers with a,d?1. Using Chebyshev?s inequalities for the function π(x) and some results concerning arithmetic progressions of prime numbers, we study the Diophantine equation

     

Diophantine equations between polynomials obeying second order recurrences

Periodica Mathematica Hungarica -     

Diophantine representations of linear recurrences. I

Direct constructions of Diophantine representations of linear recurrent sequences are discussed. These constructions generalize already known results for second-order recurrences. Some connections of this problem with the theory of units in rings of algebraic integers are shown. It is proved that the required representations exist only for second-, third-, and fourth-order sequences. In the two last-mentioned cases certain additional restrictions on their coefficients must be imposed. Bibliography:14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 52–60.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

Combinatorial numbers in binary recurrences

We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.      

Diophantine equations of Pellian type

We investigate the solutions of diophantine equations of the form dx²−d^?y²=±t for t∈{1,2,4} and their connections with ideal theory, continued fractions and Jacobi symbols.     

Existential arithmetization of Diophantine equations

A new method of coding Diophantine equations is introduced. This method allows (i) checking that a coded sequence of natural numbers is a solution of a coded equation without decoding; (ii) defining by a purely existential formula, the code of an equation equivalent to a system of indefinitely many copies of an equation represented by its code.The new method leads to a much simpler construction of a universal Diophantine equation and to the existential arithmetization of Turing machines, register machines, and partial recursive functions.     

Multidimensional system of Diophantine equations

An asymptotics for the number of solutions to a system of three Diophantine equations of additive type in six variables is found. Each additive summand of these equations is a simplest form whose degree in each variable does not exceed 1.