Andrej Dujella

A proof of the Hoggatt-Bergum conjecture

It is proved that if k and d are positive integers such that the product of any two distinct elements of the set {F2k, F2k+2, F2k+4, d} increased by 1 is a perfect square, than d has to be 4F2k+1F2k+2F2k+3. This is a generalization of the theorem of Baker and Davenport for k = 1.

Complete Solution of a Problem of Diophantus and Euler

It is proved that there does not exist a set of four positive integers with the property that the product of any two of its distinct elements plus their sum is a perfect square. This settles an old problem investigated by Diophantus and Euler.

On the size of Diophantine m-tuples

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy + n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n| ≤ 400 then |S| ≤ 32, and if |n| > 400 then |S| < 267.81 log |n|(loglog |n|) 2 . The question whether there exists an absolute bound (independent on n) for |S| still remains open.

Root separation for irreducible integer polynomials

We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).

On a problem of Diophantus for higher powers

Let

On the Exceptional Set in the Problem of Diophantus and Davenport

k\ge 3

On the family of Diophantine triples {;k-1, k+1, 16k^3-4k};

be an integer. We study the possible existence of finite sets of positive integers such that the product of any two of them increased by 1 is a

Diophantine m-tuples for linear polynomials

k

Ranks of Elliptic Curves with Prescribed Torsion over Number Fields

th power.

On the rank of elliptic curves coming from rational Diophantine triples

The Greek mathematician Diophantus of Alexandria noted that the numbers x,x + 2, 4x + 4 and 9x + 6, where x = 1/16, have the following property: the product of any two of them increased by 1 is a square of a rational number (see [4]). Fermat first found a set of four positive integers with the above property, and it was {1,3,8,120}. Later, Davenport and Baker [3] showed taht if d is a positive integer such taht the set {1,3,8,d} has the property of Diophantus, then d has to be 120.