Marija Bliznac Trebješanin

Faculty of Science, University of Split · Department of mathematics

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 eq...

Let (𝑃 𝑛 ) 𝑛≥0 and (𝑄 𝑛 ) 𝑛≥0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 ≥ 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 nonnegative integers (𝑚, 𝑛, 𝑘, 𝑏,...

Let a and b = ka be positive integers with k ∈ {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 families of positive integers c depending on a, if {a, b, c, d} is a set of positive integers which has the property that the product of any t...

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...

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

Let n≠0 be an integer. A set of m distinct 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. We give an overview of the current state of the problems related to D(4)-m-tuples. More precisely, we present some results supporting the validity of a conjecture about the regu...

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 t...

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 pro...

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.

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.

Fermatov posljednji teorem i potraga za njegovim dokazom je poznata i onima koji nisu detaljno proučavali matematiku. U ovom radu ćemo dati kratki uvod u Posljednji teorem i predstaviti konstrukciju Pitagorinih trojki i Fermatovu metodu beskonačnog spusta kojom je dokazao da je teorem istinit za slučaj n = 4, to jest, da diofantska jednadžba x**4+y...

U ovom radu predstavit će se igra RasTpad, autora Krune Matića, te dati njena matematička analiza koja opisuje strategiju pronalaska rješenja igre i osnovni pristup analizi broja rješenja s obzirom na zadano početno stanje tablice igre.

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 $D(4)$ and confirm the conjecture of uniqueness of such extension in some special cases.

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) and confirm the conjecture of uniqueness of such extension in some special cases.

In this paper we prove, under some assumptions, that every polynomial D(−1)-triple in Z[X] can only be extended to a polynomial D(−1; 1)-quadruple in Z[X] by polynomials d ±. More precisely, if {a, b, c, d} is a polynomial D(−1; 1)-quadruple, then d ± = −(a + b + c) + 2(abc ± rst), where r, s and t are polynomials from Z[X] with positive leading co...

In this paper we prove a conjecture that D(4)-quintuple does not exist using mostly the methods used in the proof of the same conjecture for D(1)-quintuples. However, some modifications were needed. Also, we give a new version of the Rickert's theorem that can be applied on some D(4)-quadruples.

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.

We improve the known upper bound for the number of Diophantine $D(4)$ -quintuples by using the most recent methods that were developed in the $D(1)$ case. More precisely, we prove that there are at most $6.8587\times 10^{29}$ $D(4)$ -quintuples.