We make several improvements to methods for finding integer solutions to x 3+ y 3+ z 3= k
for small values of k. We implemented these improvements on Charity Engine's global …

For integers k, we consider the affine cubic surface V k given by M (x)= x 1 2+ x 2 2+ x 3 2-x
1 x 2 x 3= k. We show that for almost all k the Hasse Principle holds, namely that V k (Z) is …

We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic
applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be …

In the eighties, Hooley applied the Grand Riemann Hypothesis, and what practically
amounts to the general Langlands reciprocity (automorphy) conjecture, in a fresh new way …

Research on power values of power sums has gained much attention of late, partially due to
the explosion of refinements in multiple advanced tools in (computational) number theory in …

This paper initiates a novel research direction in the theory of Diophantine equations: define
an appropriate version of the equation's size, order all polynomial Diophantine equations …

By first solving the equation $ x^ 3+ y^ 3+ z^ 3= k $ with fixed $ k $ for $ z $ and then
considering the distance to the nearest integer function of the result, we turn the sum of three …

We study the integral Brauer--Manin obstruction for affine diagonal cubic surfaces, which we
employ to construct the first counterexamples to the integral Hasse principle in this setting …

There are these two young fish swimming along and they happen to meet an older fish
swimming the other way, who nods at them and says “Morning, boys. How's the water?” And …

В этой статье продолжается обсуждение роли компьютера в современных
исследованиях по аддитивной теории чисел, в первую очередь по легкой проблеме …