arXiv subjects

Marija Bliznac Trebješanin

Publications and source records attributed to Marija Bliznac Trebješanin.

13 recordsLinked to original sources

arXiv2026

Euclidean Triangles Associated with Diophantine $D(4)$-triples

In this paper we study Euclidean triangles associated with Diophantine $D(4)$-triples. For a $D(4)$-triple $\{a,b,c\}$, $a b-a$. For a fixed shortest side $r$, we describe all triangles associated with $D(4)$-triples in terms of the factorisations of $r^2-4$, give a divisor criterion for their number, and determine their angle type. We also prove that two similar triangles associated with $D(4)$-triples are necessarily congruent. Finally, for the family of $D(4)$-triples $\{r-2,r+2,4r\}$, we obtain a Pell-equation characterisation of the Heronian triangles associated with these $D(4)$-triples.

math.NT↗

arXiv2026

On the Diophantine Equations Arising from Three-Term $b$-Concatenations in Pell-Type Sequences

We study the Diophantine problem of representing a Pell or Pell--Lucas number as the base-$b$ concatenation of three other Pell or Pell--Lucas numbers. Under the condition that the index of the middle block does not exceed that of the leading block, we obtain an explicit upper bound for the sequence indices in terms of $b$. For $2\leq b\leq15$, we determine exactly $51$ solutions up to the identification $Q_0=Q_1=2$, while there are $74$ solutions when distinct index choices are counted separately.

math.NT↗

arXiv2026

On Diophantine pairs and triples of triangular numbers

We investigate Diophantine pairs and triples of triangular numbers with the property $D(a)$ for a non-zero integer $a$. We prove that if a triangular number belongs to a $D(a)$-pair, it can be extended to infinitely many $D(a)$-triples of triangular numbers. Additionally, we determine infinite families of integers $a$ that admit such pairs, as well as families for which no $D(a)$-pairs can exist.

math.NT↗

arXiv2026

Thabit and Williams Numbers Base $b$ as a Sum or Difference of Two $g$-Repdigits

We investigate cases where Thabit and Williams numbers in base $b$ can be expressed as the sum or difference of two $g$-repdigits. For specific values of $b$ and $g$, we describe parametric solutions yielding infinitely many solutions for some equations and establish upper bounds for the parameters of the remaining finitely many solutions. As an illustration, we also provide a complete solution for some equations.

math.NT↗

arXiv2025

Extensions of $D(4)$-pairs $\{a, ka\}$ with $k\in \{7,8,10,11,12,13\}$

We study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$. Firstly, we show that it can be extended to a $D(4)$-triple with an element c, which is a member of a family of positive integers depending on a. Then, we prove that such a triple has a unique extension to a $D(4)$-quadruple.

math.NT↗

arXiv2025

On Diophantine triples containing a triangular number

A general construction yielding infinitely many families of $D(m^2)$-triples of triangular numbers is presented. Moreover, each triple obtained from this construction contains the same triangular number $T_n$.

math.NT↗

arXiv2024

On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers

Let $(P_n)_{n\ge 0}$ and $(Q_n )_{n\ge 0}$ be the Pell and Pell-Lucas sequences. Let $b$ be a positive integer such that $b\ge 2.$ In this paper, we prove that the following two Diophantine equations $P_{n}=b^{d}P_{m}+Q_{k}$ and $P_{n}=b^{d}Q_{m}+P_{k}$ with $d,$ the number of digits of $P_k$ or $Q_k$ in base $b,$ have only finitely many solutions in nonegative integers $(m, n, k, b, d)$. Also, we explicitly determine these solutions in cases $2 \le b \le 10.$

math.NT↗

arXiv2024

On extensions of $D(4)$-triples by adjoining smaller elements

This paper examines the problem of obtaining a $D(4)$-quadruple by adding a smaller element to a $D(4)$-triple. We prove some relations between elements of observed hypothetical $D(4)$-quadruples under which conjecture of the uniqueness of such an extension holds. Also, it is shown that for any $D(4)$-triple there are at most two extensions with a smaller element.

math.NT↗

arXiv2023

Polynomial $D(4)$-quadruples over Gaussian Integers

A set $\{a, b, c, d\}$ of four non-zero distinct polynomials in $\mathbb{Z}[i][X]$ is said to be a Diophantine $D(4)$-quadruple if the product of any two of its distinct elements increased by 4 is a square of some polynomial in $\mathbb{Z}[i][X]$. In this paper we prove that every $D(4)$-quadruple in $\mathbb{Z}[i][X]$ is regular, or equivalently that the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds for every $D(4)$-quadruple in $\mathbb{Z}[i][X]$.

math.NT↗

arXiv2022

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

Let $a$ and $b=ka$ 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 $\{a, ka\}.$ More precisely, we prove that by considering three families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is the set of positive integers which has the property that the product of any two of its elements increased by $4$ is a perfect square, then $d$ in given by $$d=a+b+c+\frac{1}{2}\left(abc\pm \sqrt{(ab+4)(ac+4)(bc+4)}\right).$$ As a corollary, we prove that any $D(4)$-quadruple which contains the pair $\{a, ka\}$ is regular.

math.NT↗

arXiv2022

$D(4)$-triples with two largest elements in common

In this paper we consider two new conjectures concerning $D(4)$-quadruples and prove some special cases which support their validity. The main result is a proof that $\{a,b,c\}$ and $\{a+1,b,c\}$ cannot both be $D(4)$-triples.

math.NT↗

arXiv2020

Extension of a Diophantine triple with the property $D(4)$

In this paper, we give an upper bound on the number of extensions of a triple to a quadruple for the Diophantine $m$-tuples with the property $D(4)$. We also confirm the conjecture of the uniqueness of such an extension in some special cases.

math.NT↗

arXiv2017

Nonexistence of $D(4)$-quintuples

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 applied on some $D(4)$-quadruples.

math.NT↗