We develop a variety of new techniques to treat Diophantine equations of the shape $ x^ 2+
D= y^ n $, based upon bounds for linear forms in $ p $-adic and complex logarithms, the …

Let D be a nonsquare integer, and let k be an integer with| k|≥ 1 and gcd (D, k)= 1. In the
part I of this paper, using some properties on the representation of integers by binary …

Let k be a positive integer. In this paper, using the modular approach, we prove that if k≡ 0
(mod 4), 30< k< 724 and 2 k− 1 is an odd prime power, then under the GRH, the equation x …

We describe a computationally efficient approach to resolving equations of the form C 1 x 2+
C 2= yn in coprime integers, for fixed values of C 1, C 2 subject to further conditions. We …

Perfect error-correcting codes allow for an optimal transmission of information while
guaranteeing error correction. For this reason, proving their existence has been a classical …

Given a prime number $ q $ and a squarefree integer $ C_1 $, we develop a method to
explicitly determine the tuples $(y, n,\alpha) $ for which the difference $ y^ nq^\alpha $ has …

Let c be a fixed positive integer with c> 1. Very recently, Terai et al.(Int Math Forum 17: 1–10,
2022) conjectured that the equation x 2+(4 c) y=(c+ 1) z has only one positive integer …

In this paper, we study the integer solutions of a family of Fermat‐type equations of signature
(2, 2 n, n) (2,2n,n), C x 2+ qky 2 n= zn Cx^2+q^ky^2n=z^n. We provide an algorithmically …

We answer a question of Samir Siksek, asked at the open problems session of the
conference “Rational Points 2022”, which, in a broader sense, can be viewed as a reverse …

Denote by h= h (-p) the class number of the imaginary quadratic field âš (-p) with p prime. It
is well known that h= 1 for p {3, 7, 11, 19, 43, 67, 163}. Recently, all the solutions of the …