Complementary conjecture revisited

Bloch's Conjecture Revisited

Let X be a non-singular projective complex surface. We can show that Bloch's conjecture (i.e. , that if p g = 0 then the Albanese kernel vanishes) is equivalent to the following statement:

Rank Conjecture Revisited

The rank conjecture says that rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. We prove the conjecture for elliptic curves E(K) over a number field K. As a corollary, one gets a simple estimate for the rank in terms of the length of period of a continued fraction attached to the E(K). As an illustration, we consider a family of elliptic...

Kemnitz' conjecture revisited

A conjecture of Kemnitz remained open for some 20 years: each sequence of 4n−3 lattice points in the plane has a subsequence of length nwhose centroid is a lattice point. It was solved independently by Reiher and di Fiore in the autumn of 2003.A refined and more general version of Kemnitz’ conjecture is proved in this note. The main result is about sequences of lengths between 3p−2 and 4p−3 in ...

The Mertens Conjecture Revisited

Pillai’s conjecture revisited

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3 2 1⁄4 c has, for jcj > 13; at most one solution in positive integers x and y: In fact, we show that if N and c are positive integers with NX2; then the equation jðN þ 1Þ Nj 1⁄4 c has at most one solution in positive integers x and y; unless ðN; cÞAfð2; ...