Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization

In this paper, we present several methods for the construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization method (ECM).     

On the number of Diophantine <Emphasis Type="Italic">m</Emphasis>-tuples

A set of m positive integers is called a Diophantine m-tuple if the product of any two of them is one less than a perfect square. It is known that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. On the other hand, there are infinitely many Diophantine m-tuples for m=2, 3 and 4. In this paper, we derive asymptotic estimates for the number of Diophantine pairs, triples and quadruples with elements less than given positive integer N. The author was supported by the Ministry of Science and Technology, Republic of Croatia, grants 0037110 and 037-0372781-2821.     

A proof of the Hoggatt-Bergum conjecture

It is proved that if and are positive integers such that the product of any two distinct elements of the set

increased by is a perfect square, then has to be . This is a generalization of the theorem of Baker and Davenport for .

     

On a problem of Diophantus for rationals

Let q be a nonzero rational number. We investigate for which q there are infinitely many sets consisting of five nonzero rational numbers such that the product of any two of them plus q is a square of a rational number. We show that there are infinitely many square-free such q and on assuming the Parity Conjecture for the twists of an explicitly given elliptic curve we derive that the density of such q is at least one half. For the proof we consider a related question for polynomials with integral coefficients. We prove that, up to certain admissible transformations, there is precisely one set of non-constant linear polynomials such that the product of any two of them except one combination, plus a given linear polynomial is a perfect square.     

Diophantine m-tuples for linear polynomials

In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coefficients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. This revised version was published online in June 2006 with corrections to the Cover Date.     

Complete Solution of a Problem of Diophantus and Euler

It is proved that there does not exist a set of four positiveintegers with the property that the product of any two of itsdistinct elements plus their sum is a perfect square. This settlesan old problem investigated by Diophantus and Euler.     

INDECOMPOSABILITY OF POLYNOMIALS AND RELATED DIOPHANTINE EQUATIONS

We present a new criterion for indecomposability of polynomialsover . Using the criterion, we obtain general finiteness resulton the polynomial Diophantine equation f(x) = g(y).     

Diophantine m-tuples for linear polynomials II. Equal degrees

In this paper we prove the best possible upper bounds for the number of elements in a set of polynomials with integer coefficients all having the same degree, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. Moreover, we prove that there does not exist a set of more than 12 polynomials with integer coefficients and with the property from above. This significantly improves a recent result of the first two authors with Tichy [A. Dujella, C. Fuchs, R.F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002) 21-33].