We give bounds on the number of solutions to the Diophantine equation (X+1/x)(Y +1/y) = n as n → ∞. These bounds are related to the number of solutions to congruences of the form ax+by = 1 modulo xy.

Here, we continue our work from [7] and study an inhomogeneous variant of a Diophantine equation concerning powers in products of consecutive terms of Lucas sequences. AMS Subject Classification: 11L07, 11N37, 11N60

The paper describes an automatic tuning procedure for a wide class of continuous-time systems without and with time delays. Every auto-tuning method has two basic steps. The first one is experimental and yields an estimated model. The second step consists in a design procedure of controllers. The developed auto-tuning procedure identifies a first order estimation model obtained by a biased rela...

We consider a class of differential equations, ẍ + γẋ + x = f(ωt), with p ∈ N and ω ∈ R, describing one-dimensional dissipative systems subject to a periodic forcing. For p = 1 the equation describes a resistor-inductor-varactor circuit, hence the name ‘varactor equation’. We concentrate on the limit cycle described by the trajectory with the same period as the forcing; numerically, for γ large...

The Diophantine Equation Hard Problem (DEHP) is a potential cryptographic problem on the Diophantine equation U = n i=1 Vixi. A proper implementation of DEHP would render an attacker to search for private parameters amongst the exponentially many solutions. However, an improper implementation would provide an attacker exponentially many choices to solve the DEHP. The AA β-cryptosystem is an asy...

We show how to determine the -th bit of Chaitin’s algorithmically random real number by solving instances of the halting problem. From this we then reduce the problem of determining the -th bit of to determining whether a certain Diophantine equation with two parameters, and , has solutions for an odd or an even number of values of . We also demonstrate two further examples of in number theory:...

Let b and c be fixed coprime odd positive integers with min{b, c} > 1. In this paper, a classification of all positive integer solutions (x, y, z) of the equation 2 (x) + b (y) = c (z) is given. Further, by an elementary approach, we prove that if c = b + 2, then the equation has only the positive integer solution (x, y, z) = (1,1, 1), except for (b, x, y, z) = (89,13,1, 2) and (2 (r) - 1, r + ...

Let N = pq be an RSA modulus, i.e the product of two large primes p and q. Without loss of generality, we assume that q < p. Morever, throughout this paper we assume that the primes p and q are balanced, in other words, that the bitsizes of the primes are equal so that q < p < 2q. Let e, d be the public and secret exponents satisfying ed ≡ 1 (mod φ(n)) where φ(n) = (p−1)(q−1) is the Euler totie...

have finitely or infinitely many solutions in rational integers x and y? Due to the classical theorem of Siegel (see Theorem 10.1 below), the finiteness problem for (1), and even for a more general equation F (x, y) = 0 with F (x, y) ∈ Z[x, y], is decidable (). One has to: • decompose the polynomial F (x, y) into Q-irreducible factors; • for those factors which are not Q-reducible, determine th...

In this paper we are concerned with a question that has already been answered, involving Fibonacci-type sequences and their characteristic numbers. We are only interested in primitive sequences iconsecutive pairs of terms have no common factors) and for these sequences we ask: What numbers can be the characteristic of a sequence, and given such a number, how many sequences have it? Thoro [1] ha...