# Alan FilipinUniversity of Zagreb · Department of Mathematics (GRAD)

Alan Filipin

PhD

## About

58

Publications

3,468

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435

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Citations since 2016

Introduction

My main topics of interest are Number Theory, Diophantine equations, systems of pellian equations and the problem of Diophantine m-tuples. I will be glad to send you all my papers which are not able for download here for your personal use if you are interested in my work.

Additional affiliations

March 2003 - present

Education

September 2004 - November 2006

## Publications

Publications (58)

Let b be a positive integer such that 2 ≤ b ≤ 10. In this study, we find all Pell or Pell-Lucas numbers as concatenations of two repdigits in base b. As a corollary, we show that the largest Pell or Pell-Lucas numbers which can be expressible as a concatenations of two repdigits in base b with 2 ≤ b ≤ 9 are P 11 = 5741 and Q 5 = 82, respectively.

Let k ≥ 1 be an integer and let P_k and Q_k be the k-th Pell number and k-th Pell-Lucas number, respectively. In this paper, we prove that if d is a positive integer such that {P_{2k} , P_{2k+2}, 2P_{2k+2} , d} is a Diophantine quadruple, then d = P_{2k+1} Q_{2k+1} Q_{2k+2}. We deduce that the pair {P_{2k} , 2P_{2k+2} } cannot be extended to an irr...

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...

Let k be a positive integer. In [3], it is conjectured that if {k,k+1,c,d}isaD(-k)-quadruple with c

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, \(\{k-1, k+1, 4k\}\),...

Let p be a prime such that \(4p^2+1\) is also a prime. In this paper, we prove that the \(D(-1)\)-set \(\{1,4p^2+1,1-p\}\) cannot be extended with the forth element d such that the product of any two distinct elements of the new set decreased by 1 is a square in the ring \({{\mathbb {Z}}}[\sqrt{-p}]\).

In this paper, we find all Padovan and Perrin numbers which can be expressible as a products of two repdigits in the base b with 2≤b≤10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \be...

Let \(n\ne 0\) be an integer. A set of m distinct positive integers \(\{a_1,a_2,\ldots ,a_m\}\) is called a D(n)-m-tuple if \(a_ia_j + n\) is a perfect square for all \(1\le i < j \le m\). 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 \(d=1\). The proof relies not only on...

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...

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}[...

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 $d=1$.

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 extende...

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 st...

In this paper, we study sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. It is shown that if the two smallest elements have the form \({ KA}^2\), \(4{ KA}^4 \pm 4 A\) for some positive integers A and K, and the third one is chosen canonically, then any such set consistin...

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...

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

In this paper we illustrate the use of the results from [1] proving that $D(4)$-triple $\{a, b, c\}$ with $a < b < a + 57\sqrt{a}$ has a unique extension to a quadruple with a larger element. This furthermore implies that $D(4)$-pair $\{a, b\}$ cannot be extended to a quintuple if $a < b < a + 57\sqrt{a}$.

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.

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. Let $k$ be a positive integer. By elementary means, we show that the $D(-8k^2)$-pair $\{8k^2, 8k^2+1\}$ can be extended to at most a quadruple (the third and fourth elemen...

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. Let $k$ be a positive integer. By elementary means, we show that the $D(-8k^2)$-pair $\{8k^2, 8k^2+1\}$ can be extended to at most a quadruple (the third and fourth elemen...

In these notes, we introduce definitions and theorems that are crucial for understanding and applications of linear forms in logarithms. Some Baker type inequalities that are easy to apply are introduced. In order to illustrate this very important machinery, we present some examples and show, among other things, that the largest Fibonacci number ha...

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.

We prove that if \({\{k, 4k + 4, 9k + 6, d\}}\), where \({k \in \mathbb{Z}[i]}\), \({k \neq 0, -1}\), is a Diophantine quadruple in the ring of Gaussian integers, then
$$d = 144k^3 + 240k^2 + 124k + 20.

A set of positive integers a1, a2,... , am with the property that aiaj +1 is a perfect square for all distinct indices i and j between 1 and m is called Diophantine. In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a < b < c < d < e and g = gcd(a,b), then b > 3ag; moreover, if c > a + b + 2√ab + 1 then b > max{24ag, 2 a3/2...

Let k ≥ 1 be an integer and let Fkbe the k-th Fibonacci number and Pkk-th Pell number. In this paper we prove that the pairs {K2k, F2k+6} and {P2k, P2k+4} cannot be extended to a D(4)-quintuple.

Let k ≥ 1 be an integer and let Fk be the kth Fibonacci number. In this paper we prove that if {F2k, 5F2k, c, d} with c < d is the set of four positive integers such that any product of its two distinct elements increased by 4 is a perfect square, then d is uniquely determined by k and c.

This paper is a continuation of a recent paper (see [10]), in which for a fixed Diophantine pair {a,b}{a,b} with a<ba<b, we gave an upper bound for minimal c such that {a,b,c,d}{a,b,c,d} is an irregular Diophantine quadruple with b<c<db<c<d. Here, we provide three very good examples to support the theory developed in [10].

In this paper the known upper bound 10^96 for the number of Diophantine quintuples is reduced to 5.32*10^32. The key ingredient for the improvement is that certain individual bounds on parameters are now combined with a more efficient counting of tuples, and estimated by sums over divisor functions. As a side effect, we also improve the known upper...

Let a and b be positive integers with a<b, such that ab+1 is a perfect square. In this paper we give an upper bound for the minimal positive integer c such that {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 $1$ is a perfect square and $d \neq a+b+c+2(abc\pm\sqrt{(ab+1)(a...

In this paper we prove that if k >= 3, c and d are positive integers with c < d and the set {k−2, k+2, c, d} has the property that the product of any of its distinct elements increased by 4 is a perfect square, then d is uniquely determined.

In this paper we prove that if k >= 3 and d are positive integers and the set {k − 2, k + 2, 4k^3 −4k, d} has the property that the product of any of its distinct elements increased by 4 is a perfect square, then d = 4k or d = 4k^5 − 12k^3 + 8k.

A set of m positive integers with the property that the product of any two of them increased by 4 is a perfect square is called a D(4)-m-tuple. In this paper we consider the extensibility of general D(4)-pair {a, b} and prove some results that support the conjecture that there does not exist a D(4)-quintuple.

We prove that if , c and d are positive integers with and the set has the property that the product of any of its distinct elements increased by 4 is a perfect square, then d is uniquely determined. In the proof we use the standard methods used in solving similar problems. Namely, we firstly transform our problem into solving the system of simultan...

A set of m distinct positive integers is called a Diophantine m-tuple if the product of any two of its distinct elements increased by 1 is a perfect square. It is known that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. In this paper, we prove that there are at most 10^96 Diophantine quint...

In this paper we prove that if k≥3 and d are positive integers and the set {k-2,k+2,4k 3 -4k,d} has the property that the product of any two of its distinct elements increased by 4 is a perfect square, then d=4k or d=4k 5 -12k 3 +8k.

In this paper, we show that some parametric families of D(−1)-triples cannot be extended to D(−1)-quadruples. Using this result, we further show that in each case of r = p^k, r = 2p^k, r^2 + 1 = p and r^2 + 1 = 2p^k for an odd prime p and a positive integer k, the D(−1)-pair {1, r^2 + 1} cannot be extended to a D(−1)-quadruple.

Let k be a positive integer. In this paper, we study a parametric family of the sets of integers {k,A(2)k + 4A, (A + 1)(2)k + 4(A + 1), d}. We prove that if d is a positive integer such that the product of any two distinct elements of that set increased by 4 is a perfect square, then d = (A(4) broken vertical bar 2A(3) broken vertical bar A(2))k(3)...

A set of m distinct positive integers is called a D(−1)-m-tuple if the product of any distinct two elements in the set decreased by one is a perfect square. In this paper, we show that if {1, b, c, d} with b < c < d is a D(−1)-quadruple, then c < 9.6b^4.

A D(4)-m-tuple is a set of m positive integers with the property that the product of any two of them increased by 4 is a perfect square. It is known that there does not exist a D(4)-sextuple. In this paper we show that the number of D(4)-quintuples is less than 10323. Moreover, we prove that if {a, 6, c, d, e} is a D(4)-quintuple, then max{a, b, c,...

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a prime number. In this paper we prove that the D(-k 2)-triple {1, k 2 + 1,k 2 + 4} cannot be extended to a D(-k 2)-quadruple if k ≠ 3. And for k = 3 we prove that if the set {1,10...

Let n be a nonzero integer. A set of m distinct integers is called a D(n)- m-tuple if the product of any two of them increased by n is a perfect square. Let k be a prime number. In this paper we prove that the D(-k^2)-triple {1, k^2 +1, k^2 +4} cannot be extended to a D(-k 2)-quadruple if k >= 5. And if k = 3 we prove that if the set {1, 10, 13, d}...

Let k; be a positive integer. In this paper we study the D(4)-quadruples {F 2k,F 2k+6,4F 2k+4,d}, where F k is a feth Fibonacci number. We prove that if d is a positive integer such that the product of any two distinct elements of the set increased by 4 is a perfect square, then d = 4 F2k+2F 2k+3F 2k+5. Therefore, we prove the uniqueness of the ext...

In this paper, we first show that for any fixed D(−1)-triple {1, b, c} with b < c, there exist at most two d’s such that {1, b, c, d} is a D(−1)-quadruple with c < d. Using this result, we further show that there exist at most 10^356 D(−1)-quadruples.

Let k be a positive integer. In this paper, we study the D(4)-quadruples {F_{2k}, F_{2k+6}, 4F_{2k+4}, d}, where F_k is a kth Fibonacci number. We prove that if d is a positive integer such that the product of any two distinct elements of the set increased by 4 is a perfect square, then d = 4F_{2k+2}F_{2k+3}F_{2k+5}. Therefore, we prove the uniquen...

In this paper we prove that if a positive integer d has the property that for an integer k >= 1 each of (k^12 + 1)d + 1, (k^12 + 2k^6 + 2)d + 1 and (4k^12 + 4k^6 + 5)d + 1 is a perfect square, then d = 16k^36 + 48k^30 + 100k^24 + 120k^18 + 112k^12 + 60k^6 + 24.

In this paper, we prove that there does not exist a set of 7 positive integers such that the product of any two of its distinct elements increased by 4 is a perfect square.

In this paper, we prove that if k and d are two positive integers such that the product of any two distinct elements of the set {k + 2, 4k, 9k + 6, d} increased by 4 is a perfect square, then d = 36k^3 + 96k^2 + 76k + 16.

A D(4)-m-tuple is a set of m positive integers with the property that the product of any two of them increased by 4 is a perfect square. Moreover, we call a D(4)-quadruple {a, b, c, d} such that d > max{a, b, c} regular if d = a + b + c + 1/2 (abc + rst) ; where r^2 = ab + 4, s^2 = ac + 4, t^2 = bc + 4. In this paper we prove that any D(4)-quintupl...

All complex, pure quartic fields with maximal orders generated by their units are determined. Furthermore, a quantitative version of the unit sum number problem is considered.

In this paper we use techniques based on the Schmidt's subspace theorem to obtain results on a quantitative version of the so called unit sum number problem. In particular, we solve a problem related to a recent paper of M. Jarden and W. Narkiewicz.

Let n be an integer. D(n)-m-tuple is a set of m positive integers with the property that the product of any two of them increased by n is a perfect square. In this paper, we prove that there does not exist a D(4)-sextuple.

In this paper, we prove that if {a, b, c, d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements increased by 4 is a square of a polynomial with integer coefficients, then (a + b − c − d)^2 = (ab + 4)(cd + 4).

Let n be an integer. A set of m positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. In this paper, we consider extensions of some parametric families of D(16)-triples. We prove that if {kÃ¢ÂˆÂ’4,k+4,4k,d}, for kÃ¢Â‰Â¥5, is a D(16)-quadruple, then d=k3Ã¢ÂˆÂ’4k. Furthermore, if {kÃ¢ÂˆÂ’4,4k...

The D(-1)-quadruple conjecture states that there does not ex- ist a set of four positive integers such that the product of any two distinct elements is one greater than a perfect square. An efiec- tive proof is given by showing that if {a;b;c;d} is such a set then max{a;b;c;d} < 10^10^23 , leaving open a completely determined (but currently computa...

We prove that there do not exist different positive integers c,d>1 such that the product of any two distinct elements of the set {1,10,c,d} diminished by 1 is a perfect square.