Diophantine Quadruples Research Articles

On the extensions of the Diophantine triples in Gaussian integers

A Diophantine m-tuple is a set of m distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple \(\{k-1, k+1, 16k^3-4k\}\) in Gaussian integers \({\mathbb {Z}}{[i]}\) to a Diophantine quadruple. Similar one-parameter family, \(\{k-1, k+1, 4k\}\), was studied in Franušić (Glasnik matematički 43(2):265–291, 2008), where it was shown that the extension to a Diophantine quadruple is unique (with an element \(16k^3-4k\)). The family of the triples of the same form \(\{k-1, k+1, 16k^3-4k\}\) was studied in rational integers in Bugeaud et al. (Glasg Math J 49:333–344, 2007). It appeared as a special case while solving the extensibility problem of Diophantine pair \(\{k-1, k+1\}\), in which it was not possible to use the same method as in the other cases. As authors (Bugeaud, Dujella and Mignotte) point out, the difficulty appears because the gap between \(k+1\) and \(16k^3-4k\) is not sufficiently large. We find the same difficulty here while trying to use Diophantine approximations. Then we partially solve this problem by using linear forms in logarithms.

On elliptic curves induced by rational Diophantine quadruples

In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four non-zero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/k\mathbf{Z}$ for $k=2,4,6,8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.

Formulas for Diophantine quintuples containing two pairs of conjugates in some quadratic fields

Formulas for Diophantine quintuples containing two pairs of conjugates in some quadratic fields

Diophantine quintuples containing two pairs of conjugates in some quadratic fields

Diophantine quintuples containing two pairs of conjugates in some quadratic fields

There are no Diophantine quadruples of Pell numbers

In this paper, we prove that there are no positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] such that [Formula: see text] is a Diophantine quadruple, where for a positive integer [Formula: see text], [Formula: see text] is the [Formula: see text]th Pell number.

The extensibility of the Diophantine triple {2, b, c}

Abstract The aim of this paper is to consider the extensibility of the Diophantine triple {2, b, c}, where 2 < b < c, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of c’s (depending on b). As corollary, for example, we prove that for b/2 − 1 prime, all Diophantine quadruples {2, b, c, d} with 2 < b < c < d are regular.

On the Diophantine pair {a,3a}

The purpose of the present paper is to consider the extensibility of the Diophantine triple {a,b,c}, where a<b<c with b=3a, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of c's (depending on a). As corollary, we prove that any Diophantine quadruple which contains the pair {a,3a} is regular. Finally in this paper, we will see that by considering the case b=8a we obviously obtain similar results.

Diophantine m-tuples in finite fields and modular forms

For a prime p, a Diophantine m-tuple in $$\mathbb {F}_p$$ is a set of m nonzero elements of $$\mathbb {F}_p$$ with the property that the product of any two of its distinct elements is one less than a square. In this paper, we present formulas for the number $$N^{(m)}(p)$$ of Diophantine m-tuples in $$\mathbb {F}_p$$ for $$m=2,3$$ and 4. Fourier coefficients of certain modular forms appear in the formula for the number of Diophantine quadruples. We prove that asymptotically $$N^{(m)}(p)=\displaystyle \frac{1}{2^{m \atopwithdelims ()2 }}\frac{p^m}{m!} + o(p^m)$$ , and also show that if $$p>2^{2m-2}m^2$$ , then there is at least one Diophantine m-tuple in $$\mathbb {F}_p$$ .

Identities for linear recursive sequences of order &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ 2 $&lt;/tex-math&gt;&lt;/inline-formula&gt;

&lt;p style='text-indent:20px;'&gt;We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [&lt;xref ref-type="bibr" rid="b16"&gt;16&lt;/xref&gt;]. Numerous identities are constructed, and many well known identities can be proved readily by using this unified rule. Various Catalan-like and Cassini-like identities are given for recursive number sequences and recursive polynomial sequences. Sets of identities for Diophantine quadruple are shown.&lt;/p&gt;

Doubly regular Diophantine quadruples

For a nonzero integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine m-tuples and certain family of elliptic curves, we show that there are infinitely many essentially different sets consisting of perfect squares which are simultaneously D(n_1)-quadruples and D(n_2)-quadruples with distinct non-zero squares n_1 and n_2.

Diophantine quadruples in $$\mathbb {Z}[i][X]$$

In this paper, we prove that every Diophantine quadruple in $$\mathbb {Z}[i][X]$$ is regular. More precisely, we prove that if $$\{a, b, c, d\}$$ is a set of four non-zero polynomials from $$\mathbb {Z}[i][X]$$ , not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from $$\mathbb {Z}[i][X]$$ , then $$\begin{aligned} (a+b-c-d)^2=4(ab+1)(cd+1). \end{aligned}$$

Diophantine triples with largest two elements in common

In this paper we prove that if $$\{a,b,c\}$$ is a Diophantine triple with $$a<b<c$$ , then $$\{a+1,b,c\}$$ cannot be a Diophantine triple. Moreover, we show that if $$\{a_1,b,c\}$$ and $$\{a_2,b,c\}$$ are Diophantine triples with $$a_1<a_2<b<c < 16b^3$$ , then $$\{a_1,a_2,b,c\}$$ is a Diophantine quadruple. In view of these results, we conjecture that if $$\{a_1,b,c\}$$ and $$\{a_2,b,c\}$$ are Diophantine triples with $$a_1<a_2<b<c$$ , then $$\{a_1,a_2,b,c\}$$ is a Diophantine quadruple.

CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES

Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.

Coincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Triples

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by 1 is a perfect square. A conjecture in this field asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple in some sense. This problem is reduced to study the coincidence between certain two binary recurrent sequences of integers. As an analogy of this, we consider a similar coincidence on the polynomial ring in one variable over the integers, and determine it completely. Our result is regarded as a generalization of a result in the paper “Complete solution of the polynomial version of a problem of Diophantus” by A. Dujella, C. Fuchs in Journal of Number Theory 106 (2004) on the polynomial variant of Diophantine tuples.

A Certain Type of Regular Diophantine Triples and Their Non-Extendability

In the present paper, we consider some D(s) Diophantine triples for a prime integer s with its negative case/value although there exist infinitely many Diophantine triples. We give several properties of such Diophantine triples and prove that they are non – extendability to D(s) Diophantine quadruple using algebraic and elementary number theory structures.

Diophantine pairs that induce certain Diophantine triples

Diophantine tuples are sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. In the main theorem of this paper it is shown that any Diophantine triple, the second largest element of which is between the square and four times the square of the smallest one, is uniquely extended to a Diophantine quadruple by joining an element exceeding the largest element in the triple. A similar result is obtained under the hypothesis that the two smallest elements have the form T2+2T, 4T4+8T3−4T for some positive integer T, which we encounter as an exceptional case. The main theorem implies that the same is valid for triples with smallest elements KA2, 4KA4±4A for some positive integers A and K∈{1,2,3,4}.

On the size of Diophantine m-tuples in imaginary quadratic number rings

A Diophantine [Formula: see text]-tuple is a set of [Formula: see text] distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that [Formula: see text]. Our proof is relatively simple compared to the proofs of similar results in positive integers.

Rational Diophantine sextuples with square denominators

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely many rational Diophantine sextuples such that the denominators of all the elements in the sextuples are perfect squares.

A polynomial variant of a problem of Diophantus and its consequences

In this paper we prove that every Diophantine quadruple in R[X] is regular. In other words, we prove that if {;a, b, c, d}; is a set of four non-zero elements of R[X], not all constant, such that the product of any two of its distinct elements increased by 1 is a square of an element of R[X], then (a+b−c−d)^2=4(ab+1)(cd+1). Some consequences of the above result are that for an arbitrary n ∈N there does not exist a set of five non-zero elements from Z[X], which are not all constant, such that the product of any two of its distinct elements increased by n is a square of an element of Z[X]. Furthermore, there can exist such a set of four non-zero elements of Z[X] if and only if n is a square.

Some family of Diophantine pairs with Fibonacci numbers

A set {a1, a2, …, am} of positive integers is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all 1 ≤ i < j ≤ m. In this paper, we prove that if {F2k, F2k+4, c} is a Diophantine triple then there is a unique positive integer d such that c < d and {F2k, F2k+4, c, d} is a Diophantine quadruple.