This paper is devoted to the generalized Fermat equation xp+ yq= zr, where p, q and r are
integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent …

Generalised Fermat equation (GFE) is the equation of the form $ ax^ p+ by^ q= cz^ r $,
where $ a, b, c, p, q, r $ are positive integers. If $1/p+ 1/q+ 1/r< 1$, GFE is known to have at …

Wiles' proof of Fermat's last theorem initiated a powerful new approach towards the
resolution of certain Diophantine equations over $\mathbb {Q} $. Numerous novel obstacles …

In this paper, we give a resolution of the generalized Fermat equations\begin {equation*} x^
5+ y^ 5= 3 z^ n\quad\text {and}\quad x^{13}+ y^{13}= 3 z^ n,\end {equation*} for all integers …

Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^ 3+ x^
3+(x+ 1)^ 3= z^ 2$ in integers $ x $, $ z $. The generalization $(x-1)^ k+ x^ k+ (x+ 1)^ k= z^ n …

On the sum of fourth powers in arithmetic progression Page 1 International Journal of Number
Theory Vol. 17, No. 1 (2021) 191–221 c World Scientific Publishing Company DOI …

In this paper, we consider the equation x ²–q 2 k+ 1= yn, q∤ x, 2| y, for integers x, q, k, y and
n, with k≥ 0 and n≥ 3. We extend work of the first and third-named authors by finding all …

Abstract The Lebesgue-Nagell equation, x²+ D= yⁿ, n≥ 3 integer, is a classical family of
Diophantine equations that has been extensively studied for decades. Bugeaud, Mignotte …

The main topic of this thesis is the study of some variants of Catalan's equation, the
exponential Diophantine equation (an− 1)(bn− 1)= z2, where a, b are fixed integers and …

Let x, y, z, n, α∈ ℤ with α≥ 1, p and n≥ 5 primes. In 2011, Michael Bennett, Florian Luca
and Jamie Mulholland showed that the equation involving a twisted sum of cubes [equation …