We explore the prospects of a monist account of explanation for both non-causal
explanations in science and pure mathematics. Our starting point is the counterfactual theory …

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 …

For any fixed relatively prime positive integers $ a $, $ b $ and $ c $ with $\min\{a, b, c\}> 1$,
we prove that the equation $ a^ x+ b^ y= c^ z $ has at most two solutions in positive integers …

Let $ K $ be a totally real number field with narrow class number one and $ O_K $ be its ring
of integers. We prove that there is a constant $ B_K $ depending only on $ K $ such that for …

We obtain additional Diophantine applications of the methods surrounding Darmon's
program for the generalized Fermat equation developed in the first part of this series of …

Text We present a general algorithm for solving all two-variable polynomial Diophantine
equations consisting of three monomials. Before this work, even the existence of an …

We construct a local deformation problem for residual Galois representations $\bar {\rho} $
valued in an arbitrary reductive group $\hat {G} $ which we use to develop a variant of the …

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 …

For any fixed coprime positive integers $ a, b $ and $ c $ with $\min\{a, b, c\}> 1$, we prove
that the equation $ a^ x+ b^ y= c^ z $ has at most two solutions in positive integers $ x, y …

Let $ X $ be a variety over $\mathbb Q $, and let $ M $ be a (rational) moduli space over
$\mathbb Q $ of abelian varieties with $ dim (M)> dim (X) $. We study $ X (\mathbb Q) $ by …