Pure powers in recurrence sequences and some related diophantine equations

<p>Let f(n)=1 if n=1, 2^(2^(n-2)) if n \\in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \\in {6,7,8,...}. We conjecture that if a system T \\subseteq {x_i+1=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \\leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite. We show that the conjecture and Matiyasevich's conjecture on finite-fold Diophantine representations are jointly inconsistent.</p>

We find 48 more solutions to a Diophantine equation investigated by Martin Davis. Before our work, only two solutions were known. Construction of the new solutions required the factorization of several large integers. Because the equation relates to Hilbert’s Tenth Problem it is desirable to know if it has only finitely many solutions. An elaborate argument is given for the conjecture that the equation has infinitely many solutions in integers.

Perfect powers in second order linear recurrences

The Undecidability of Exponential Diophantine Equations

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$ for all integers $n\geq k$ , where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $ , where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$ , then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$ .

Let N = pq be an RSA modulus and e be a public exponent. Let j(N) = (p − 1)(q − 1) be the Euler’s totient function. The equation ex2 −j(N)y2 = z has infinitely many solutions in integers (x, y, z). We show that if x, y and z are suitably small, then one can factor the RSA modulus. Our bounds on the size of the solutions x, y, and z improve the existing bounds of some attacks on RSA such as Wiener’s continued fractions based attack, and Blömer-May’s lattice reduction based attack.

has only finitely many solutions in integers x and yv. In the first part of this paper we shall establish upper bounds for the number of solutions of (1) in coprime integers x and y under the assumption that the discriminant D(F) of F is nonzero. For most integers h these bounds improve upon those obtained by Bombieri and Schmidt in [5]. In the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of Nagell [30], Ore [32], Sandor [33], and Huxley [19]. In fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is, in general, best possible. In the second part we shall address the problem of finding forms F for which (1) has many solutions for arbitrarily large integers h. Finally we shall obtain upper bounds for the number of solutions of certain Thue-Mahler and Ramanujan-Nagell equations by appealing to estimates of Evertse, Gy6ry, Stewart, and Tijdeman [17] for the number of solutions of S-unit equations.

Tropical recurrent sequences

We study solvability of the Diophantine equationn2n=∑i=1kai2ai, in integers n,k,a1,…,ak satisfying the conditions k≥2 and ai<ai+1 for i=1,…,k−1. The above Diophantine equation (of polynomial-exponential type) was mentioned in the monograph of Erdős and Graham, where several questions were stated. Some of these questions were already answered by Borwein and Loring. We extend their work and investigate other aspects of Erdős and Graham equation. First of all, we obtain the upper bound for the value ak given in terms of k only. This mean, that with fixed k our equation has only finitely many solutions in n,a1,…,ak. Moreover, we construct an infinite set K, such that for each k∈K, the considered equation has at least five solutions. As an application of our findings we enumerate all solutions of the equation for k≤8. Moreover, by applying greedy algorithm, we extend Borwein and Loring calculations and check that for each n≤104 there is a value of k such that the considered equation has a solution in integers n+1=a1<a2<…<ak. Based on our numerical calculations we formulate some further questions and conjectures.

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of $d<0$. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond.

Classes of integer Abaffy–Broyden–Spedicato (ABS) methods have recently been introduced for solving linear systems of Diophantine equations. Each method provides the general integer solution of the system by computing an integer solution and an integer matrix, named Abaffian, with rows generating the integer null space of the coefficient matrix. The Smith normal form of a general rectangular integer matrix is a diagonal matrix, obtained by elementary nonsingular (unimodular) operations. Here, we present a class of algorithms for computing the Smith normal form of an integer matrix. In doing this, we propose new ideas to develop a new class of extended integer ABS algorithms generating an integer basis for the integer null space of the matrix. For the Smith normal form, having the need to solve the quadratic Diophantine equation, we present two algorithms for solving such equations. The first algorithm makes use of a special integer basis for the row space of the matrix, and the second one, with the intention of controlling the growth of intermediate results and making use of our given conjecture, is based on a recently proposed integer ABS algorithm. Finally, we report some numerical results on randomly generated test problems showing a better performance of the second algorithm in controlling the size of the solution. We also report the results obtained by our proposed algorithm on the Smith normal form and compare them with the ones obtained using Maple, observing a more balanced distribution of the intermediate components obtained by our algorithm.

We show that the equation in the title (with m the mth cyclotomic polynomial) has no integer solution with n  1 in the cases (m; p) = (15; 41), (15; 5581), (10; 271). These equations arise in a recent group theoretical investigation by Akhlaghi, Khosravi and Khatami

Let $q$ be an odd prime. Let $c&gt;1$ and $t$ be positive integers such that $q^{t}+1=2c^{2}$. Using elementary method and a result due to Ljunggren concerning the Diophantine equation $\\frac{x^{n}-1}{x-1}= y^{2}$, we show that the Diophantine equation $x^{2}+q^{m}=c^{2n}$ has the only positive integer solution $(x, m, n)=(c^{2}-1, t, 2)$. As applications of this result some new results on the Diophantine equation $x^{2}+q^{m} = c^{n}$ and the Diophantine equation $x^{2}+(2c-1)^{m} = c^{n}$ are obtained. In particular, we prove that Terai’s conjecture is true for $c=12,24$. Combining this result with Terai’s results we conclude that Terai’s conjecture is true for $2 \\leq c \\leq 30$.

Some time ago we extended our monogenity investigations and calculations of\ngenerators of power integral bases to the relative case. Up to now we\nconsidered (usually totally real) extensions of complex quartic fields. In the\npresent paper we consider power integral bases in relative extensions of\ntotally real fields. Totally complex quartic extensions of totally real number\nfields seems to be the most simple case, that we detail here. As we shall see,\neven in this case we have to overcome several unexpected difficulties, which we\ncan, however solve by properly (but not trivially) adjusting standard methods.\nWe demonstrate our general algorithm on an explicit example. We describe how\nthe general methods for solving relative index form equations in quartic\nrelative extensions are modified in this case. As a byproduct we show that\nrelative Thue equations in totally complex extensions of totally real fields\ncan only have small solutions, and we construct a special method for the\nenumeration of small solutions of special unit equations. These statements can\nbe applied to other diophantine problems, as well.\n

This paper proves that for all positive integers n, the equation $$ \frac{x}{y}+p\frac{y}{z}+\frac{z}{w}+p\frac{w}{x}=8np, $$ where p = 1 or p is a prime congruent to 1 (mod8), does not have solutions in positive integers.