Diophantine quadruples in Z [ √ − 2 ]

نویسندگان

  • Andrej Dujella
  • Ivan Soldo
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.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

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...

متن کامل

Adjugates of Diophantine Quadruples

Philip Gibbs Diophantine m-tuples with property D(n), for n an integer, are sets of m positive integers such that the product of any two of them plus n is a square. Triples and quadruples with this property can be classed as regular or irregular according to whether they satisfy certain polynomial identities. Given any such m-tuple, a symmetric integer matrix can be formed with the elements of ...

متن کامل

On the Exceptional Set in the Problem of Diophantus and Davenport

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 that if d is a posi...

متن کامل

On Diophantine Quadruples of Fibonacci Numbers

We show that there are only finitely many Diophantine quadruples, that is, sets of four positive integers {a1, a2, a3, a4} such that aiaj +1 is a square for all 1 ≤ i < j ≤ 4, consisting of Fibonacci numbers.

متن کامل

On the number of Diophantine m-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 extimates for the numb...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2009