Diophantine quadruples in Z [ √ − 2 ]
نویسندگان
چکیده
In this paper, we study the existence of Diophantine quadruples with the property D(z) in the ring Z[ √−2 ]. We find several new polynomial formulas for Diophantine quadruples with the property D(a+ b √−2 ), for integers a and b satisfying certain congruence conditions. These formulas, together with previous results on this subject by Abu Muriefah, Al-Rashed and Franušić, allow us to almost completely characterize elements z of Z[ √−2 ] for which a Diophantine quadruple with the property D(z) exists.
منابع مشابه
Some diophantine quadruples in the ring Z [ √ − 2 ]
A complex diophantine quadruple with the property D (z), where z ∈ Z[√−2], is a subset of Z[√−2] of four elements such that the product of its any two distinct elements increased by z is a perfect square in Z[ √−2]. In the present paper we prove that if b is an odd integer, then there does not exist a diophantine quadruple with the property D(a + b √−2). For z = a + b√−2, where b is even, we pr...
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