#### Glasnik Matematicki, Vol. 32, No.1 (1997), 1-10.

### THE PROBLEM OF DIOPHANTUS AND DAVENPORT FOR
GAUSSIAN INTEGERS

### Andrej Dujella

Department of Mathematics, University of Zagreb, Bijenicka cesta 30,
10000 Zagreb, Croatia

*e-mail:* `duje@math.hr`

**Abstract.** 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 Gaussain 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
2*i*,
4*i* },
then there exists at least one quadruple with the property
*D*(*z*).

**1991 Mathematics Subject Classification.**
11D09.

**Key words and phrases.** Diophantine quadruple,
property of Diophantus, Gaussian integers, Pell equation.

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