Department of Mathematics

A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. Diophantus found a set of four positive rationals with the above property. The first Diophantine quadruple was found by Fermat (the set {1, 3, 8, 120}). Baker and Davenport proved that this particular quadruple cannot be extended to a Diophantine quintuple.

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any two distinct elements of the set {k − 1, k + 1, 4k, d} increased by 1 is a perfect square, than d has to be 16k 3 −4k. This is a generalization of the well-known result of Davenport and Baker for k = 2.

Extending the classical Legendre's result, we describe all solutions of the inequality |α − a/b| < c/b 2 in terms of convergents of continued fraction expansion of α. Namely, we show that a/b = (rpm+1 ± spm)/(rqm+1 ± sqm) for some nonnegative integers m, r, s such that rs < 2c. As an application of this result, we describe a modification of Verheul and van Tilborg variant of Wiener's attack on RSA cryptosystem with small secret exponent. 0 2000 Mathematics Subject Classification. 11A55, 94A60.

A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. We prove that if {a, b, c} is a Diophantine triple such that b > 4a and c > max{b 13 , 10 20 } or c > max{b 5 , 10 1029 }, then there is unique positive integer d such that d > c and {a, b, c, d} is a Diophantine quadruple. Furthermore, we prove that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples.

A set of Gaussian integers is said to have the property D(z) if the product of its any two distinct elements increased by z is a square of a Gaussian integer. In this paper it is proved that if a Gaussian integer z is not representable as a difference of the squares of two Gaussian integers, then there does not exist a quadruple with the property D(z), but if z is representable as a difference of two squares and if z ∈ {±2, ±1 ± 2i, ±4i}, then there exists at least one quadruple with the property D(z).

Diophantus found four positive rational numbers 1 16 , 33 16 , 17 4 , 105 16 with the property that the product of any two of them increased by 1 is a perfect square. The first set of four positive integers with the above property was found by Fermat and that set was {1, 3, 8, 120} (see ). These two examples motivate the following definition.

The D(−1)-quadruple conjecture states that there does not exist a set of four positive integers such that the product of any two distinct elements is one greater than a perfect square. It is proved that if {a, b, c, d} is such a set, then max{a, b, c, d} < 10 10 23 , hence a proof of the conjecture is given up to a completely determined, finite (but currently computationally infeasible large) set of possible exceptions to check.