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Publications (10)

math.NT2022

On the $D (4)$ -pairs ${a, k a}$ with $k \in {2, 3, 6}$

KouÃ¨ssi Norbert AdÃ©dji, Marija Bliznac TrebjeÅ¡anin, Alan Filipin +1

Let $a$ and $b = k a$ be positive integers with $k \in {2, 3, 6},$ such that $ab + 4$ is a perfect square. In this paper, we study the extensibility of the $D (4)$ -pairs ${a, k a} .$ Mo…

math.NT2022

Pell or Pell-Lucas numbers as concatenations of two repdigits in base $b$

Kouessi Norbert Adedji, Alan Filipin, Salah Eddine Rihane +1

Let $b$ be a positive integer such that $2 \leq b \leq 10$ . In this study, we find all Pell or Pell-Lucas numbers as concatenations of two repdigits in base $b$ . As a corollary, it…

math.NT2017

Nonexistence of $D (4)$ -quintuples

Marija Bliznac TrebjeÅ¡anin, Alan Filipin

In this paper we prove a conjecture that $D (4)$ -quintuple does not exist using both classical and new methods. Also, we give a new version of the Rickert's theorem that can be appl…

math.NT2022

Fibonacci and Lucas numbers as products of three repdigits in base $g$

Kouessi Norbert Adedji, Alan Filipin, Alain Togbe

Recall that repdigit in base $g$ is a positive integer that has only one digit in its base $g$ expansion, i.e. a number of the form $a (g^{m} - 1) / (g - 1)$ , for some positive integers $m\…

math.NT2017

The extension of some D(4)-pairs

Alan Filipin

In this paper we illustrate the use of the results from [1] proving that $D (4)$ -triple ${a, b, c}$ with $a < b < a + 57 a$ has a unique extension to a quadruple with a larg…

math.NT2020

The extension of the $D (- k)$ -pair ${k, k + 1}$ to a quadruple

Nikola AdÅ¾aga, Alan Filipin, Yasutsugu Fujita

Let $k$ be a positive integer. In this paper, we prove that if ${k, k + 1, c, d}$ is a $D (- k)$ -quadruple with $c > 1$ , then $d = 1$ .

math.NT2019

On the extensions of the Diophantine triples in Gaussian integers

Nikola AdÅ¾aga, Alan Filipin, Zrinka FranuÅ¡iÄ

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…

math.NT2026

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

Nikola AdÅ¾aga, Alan Filipin, Ana JurasiÄ

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 Diophan…

math.NT2016

On the extension of $D (- 8 k^{2})$ -pair ${8 k^{2}, 8 k^{2} + 1}$

Nikola AdÅ¾aga, Alan Filipin

Let $n$ be a nonzero integer. A set of $m$ positive integers is called a $D (n)$ - $m$ -tuple if the product of any two of its distinct elements increased by $n$ is a perfect square. L…

math.NT2017

A polynomial variant of a problem of Diophantus and its consequences

Alan Filipin, Ana JurasiÄ

We prove that every Diophantine quadruple in $R [X]$ is regular. More precisely, we prove that if ${a, b, c, d}$ is a set of four non-zero polynomials from $\mathbb{R}[X]…

Publications (10)

On the D(4)-pairs {a,ka} with k∈{2,3,6}

Pell or Pell-Lucas numbers as concatenations of two repdigits in base b

Nonexistence of D(4)-quintuples

Fibonacci and Lucas numbers as products of three repdigits in base g

The extension of some D(4)-pairs

The extension of the D(−k)-pair {k,k+1} to a quadruple

On the extensions of the Diophantine triples in Gaussian integers

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

On the extension of D(−8k2)-pair {8k2,8k2+1}

A polynomial variant of a problem of Diophantus and its consequences

On the $D (4)$ -pairs ${a, k a}$ with $k \in {2, 3, 6}$

Pell or Pell-Lucas numbers as concatenations of two repdigits in base $b$

Nonexistence of $D (4)$ -quintuples

Fibonacci and Lucas numbers as products of three repdigits in base $g$

The extension of the $D (- k)$ -pair ${k, k + 1}$ to a quadruple

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

On the extension of $D (- 8 k^{2})$ -pair ${8 k^{2}, 8 k^{2} + 1}$