We construct a (smooth, projective) surface over the field of rational numbers, which is a
counterexample to the Hasse principle not accounted for by the Manin obstruction. The …

Number theory is a branch of mathematics that is primarily focused on the study of positive
integers, or natural numbers, and their properties such as divisibility, prime factorization, or …

This is the second in a series of papers where we combine the classical approach to
exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a …

We give a new algorithm using linear approximation and lattice reduction to efficiently
calculate all rational points of small height near a given plane curve C. For instance, when C …

Bibliography Page 1 Bibliography [1] Abel-Hollinger, CS, Zimmer, HG: Torsion groups of elliptic
curves with integral j-invariant over multiquadratic fields. In: Proceedings of the Int. Conf …

We consider models for genus-one curves of degree n for n= 2, 3 and 4, which arise in
explicit n-descent on elliptic curves. We prove theorems on the existence of minimal models …

In the first part we construct algorithms (over $\mathbb {Q} $) which we apply to solve $ S $-
unit, Mordell, cubic Thue, cubic Thue–Mahler and generalized Ramanujan–Nagell …

In 1842 the Belgian mathematician Eugène Charles Catalan asked whether 8 and 9 are the
only consecutive pure powers of nonzero integers. One hundred and sixty years after, the …

If $ b $ and $ d $ are given positive integers with $ b> 1$, then we show that the equation of
the title possesses at most one solution in positive integers $ X, Y $. Moreover, we give an …

Let the elliptic curve E be defined by the equationformula herewith a1,…, a6∈ ℤ. Define a
finite set of places S={q1,…, qs− 1, qs=∞} of ℚ and put Q= max {q1,…, qs− 1}. Let E (ℚ) …