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The purpose of the present article is to prove that the Diophantine equation n^x + 13^y = z^2 has exactly one solution (n, x, y, z) = (2, 3, 0, 3) where x, y and z are non-negative integers and n is a positive integer with n ≡ 2 (mod 39) and n + 1 is not a square number. In particular, (3, 0, 3) is a unique solution (x, y, z) in non-negative integers of the Diophantine equation 2^x + 13^y = z^2

In his survey “Beyond graph energy: Norms of graphs and matrices” [23] , Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph, respectively. We show that these graphs have at most two nonzero distinct singular values and investigate the proposed problems organizing our study according to the type of spectrum they can have. In most cases all graphs are characterized. Infinite families of graphs are given otherwise. We also show that all graphs satisfying the properties required in the problems are integral, except for complete bipartite graphs K p , q and disconnected graphs with a component K p , q , where pq is not a perfect square.

The balancing number $n$ and the balancer $r$ are solution of the Diophantine equation $$1+2+\cdots+(n-1) = (n+1)+(n+2)+\cdots+(n+r). $$ It is well known that if $n$ is balancing number, then $8n^2 + 1$ is a perfect square and its positive square root is called a Lucas-balancing number. For an integer $k\geq 2$, let $(F_n^{(k)})_n$ be the $k$-generalized Fibonacci sequence which starts with $0,\ldots,0,1,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. The purpose of this paper is to show that 1, 6930 are the only balancing numbers and 1, 3 are the only Lucas-balancing numbers which are a term of $k$-generalized Fibonacci sequence. This generalizes the result from [Fibonacci Quart. 2004, 42 (4), 330-340].

The Nagata Conjecture is one of the most intriguing open problems in the area of curves in the plane. It is easily stated. Namely, it predicts that the smallest degree d of a plane curve passing through r ≥ 10 general points in the projective plane P 2 with multiplicities at least l at every point, satisfies the inequality d > √ r · l. This conjecture has been proven by M. Nagata in 1959, if r is a perfect square greater than 9. Up to now, it remains open for every non-square r ≥ 10, after more than a half century of attention by many researchers. In this paper, we formulate new transcendental versions of this conjecture coming from pluripotential theory and which are equivalent to a version in C n of the Nagata Conjecture.

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

For an element $r$ of a ring $R$, a Diophantine $D(r)$ $m$-tuple is an $m$-tuple $(a_1,a_2,\ldots,a_m)$ of elements of $R$ such that for all $i,j$ with $i\neq j$, $a_ia_j+r$ is a perfect square in $R$. In this article, we compute and estimate the measures of the sets of $D(r)$ $m$-tuples in the ring $\mathbb{Z}_p$ of $p$-adic integers, as well as its residue field $\mathbb{F}_p$., 22 pages. Comments welcome

In 1927, Artin hypothesized that for any given non-zero integer a other than 1, −1, or a perfect square, there exists infinitely many primes p for which a is a primitive root modulo p. In 1967, Hooley proved it under the assumption of the generalized Riemann hypothesis. Since then, there are many analogues and generalization of this conjecture. In this paper, we work on its generalization to composite moduli in the function fields setting. Let A = F q [ t ] be the ring of polynomials over the finite field F q and 0 ≠ a ∈ A . Let C be the A-Carlitz module. Let a be a fixed element in A. For n ∈ A , C ( A / n A ) is a finite A-module. The set of all annihilators of C ( A / n A ) is an ideal and generated by a monic polynomial, denoted by λ ( n ) . Similarly, The set of all annihilators of the submodule of C ( A / n A ) generated by a is an ideal and let l a ( n ) be its monic generator. We say that a is a primitive root of n, if λ ( n ) = l a ( n ) . Define N a ( x ) : = | { n ∈ A | deg ⁡ n = x , n is monic , a is a primitive root of n } | We prove that for a given non-constant a ∈ A , a ∉ E , an exceptional set, there exists an unbounded set V of integers such that lim inf x ∈ V N a ( x ) / q x = 0 This result is analogous to Li's theorem for Artin's conjecture on composite moduli. It is the first time that this kind of results holds in the setting of the function fields.

A set of m distinct positive integers $$\{a_{1} , \ldots a_{m}\}$$ is called a $$D(q)-m$$ -tuple for nonzero integer q if the product of any two increased by q, $$a_{i}a_{j}+q, i \neq j$$ is a perfect square. Due to certain properties of the sequence, there are many D(q)-Diophantine triples related to the Fibonacci numbers. A result of Bacic and Filipin characterizes the solutions of Pellian equations that correspond to D(4)-Diophantine triples of a certain form. We generalize this result in order to characterize the solutions of Pellian equations that correspond to D(l2)-Diophantine triples satisfying particular divisibility conditions. Subsequently, we employ this result and bounds on linear forms in logarithms of algebraic numbers in order to classify all D(9) and D(64)-Diophantine triples of the form $$\{F_{2n+8},9F_{2n+4},F_{k}\}$$ and $$\{F_{2n+12},16F_{2n+6},F_{k}\}$$ , where $$F_{i}$$ denotes the ith Fibonacci number.

Let $$a>1,b$$ be two positive integers where the square-free part of b is 2pq with p, q two distinct odd primes. Recently, Cipu (Proc Am Math Soc 146:983–992, 2018) proved that if one of the following conditions holds: (i) $$2a^2-1$$ is not a perfect square, (ii) $$\{p\pmod 8,q\pmod 8\}\ne \{1,3\}$$ , then the equations $$\begin{aligned} x^2-(a^2-1)y^2=1 \quad \text {and} \quad y^2-bz^2=1 \end{aligned}$$ have solutions in positive integers if and only if $$8a^2(2a^2-1)/b$$ is a perfect square. When it exists, the solution is $$(x,y,z)=(4a^3-3a,4a^2-1, \sqrt{8a^2(2a^2-1)/b})$$ . In this paper, we completely solve these equations when the square-free part of b is 2pq.

We propose an alternative formula for solving the biquadratic equation x4+bx2+c=0, which is more convenient than the classical one whenever c is a perfect square. The method is suitable to be intro...

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.

The monopole-dimer model introduced recently is an exactly-solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a perfect square. We give a combinatorial interpretation of the square-root of the partition function for such graphs in terms of a monopole-dimer model on a new kind of graph with two types of edges which we call a dicot. The partition function of the latter can be written as a determinant, this time of a complex adjacency matrix. This formulation generalises T. T. Wu's assignment of imaginary orientation for the grid graph to planar dicots. As an application, we compute the partition function for a family of non-planar dicots with positive weights., Comment: 21 pages, 7 figures, minor improvements, final version

Penyelesaian beberapa masalah matematika tidak hanya menggunakan metode analitik tetapi juga dengan metode numerik yang prosesnya iteratif. Dalam metode numerik, selain mahasiswa memahami perhitungan tiap iterasi secara manual juga perlu untuk memahami penyelesaian menggunakan coding atau pemograman. Satu diantara program yang mudah diunduh adalah QBasic. Tulisan ini membahas penentuan akar bilangan bulat yang bukan bilangan kuadrat sempurna yaitu akar kuadrat dari 3 menggunakan metode numerik dengan teknik iteratif. Tujuan penulisan adalah melihat karakteristik komputasi dalam jumlah iterasi pada metode Heron, bagi dua, posisi palsu, dan Newton Raphson. Penelitian melalui simulasi menggunakan pemgroman QBasic. Nilai kesalahan (eror) dalam simulasi pada metode yang dilakukan ditetapkan pada nilai 1x10-10 atau iterasi masih akan jika nlai kesalahan lebih dari nilai kesalahan yang ditetapkan. Hasil simulasi didapatkan bahwa pada nilai kesalahan tersebut nilai akar kuadrat bilangan 3 berkisar pada 1,73205. Metode Heron dan metode Newton Raphson memiliki jumlah iterasi yang sama, yaitu 6 iterasi, relatif lebih sedikit dibandingkan metode iteratif lain. Kesimpulan penelitian adalah pada penentuan akar kudrat dari bilangan 3 metode Heron dan metode Newton Raphson memberikan kinerja komputasi yang lebih baik dari jumlah iterasi dibanding pada metode bagi dua dan metode posisi palsu.

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence $\{F_n\}_{n=0}^\infty$ misses a congruence class modulo every prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes., 13 pages

Recurrence relationships allow us to accurately determine logarithms, square roots, factorials, several sequences (like Fibonacci), roots of equations and more. This paper aims to describe a recurrence relationship and present some applications. It presents the concept of reciprocal matrix and allowed the approximation of non perfect square roots of positive integers and their representation in continuous fraction. In addition, there was a correspondence between this study and the so-called Fibonacci polynomials. As it is a new job, more studies are needed to expand its concepts and applications.

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 $d=1$., 15 pages

We study some counting questions concerning products of positive integers $u_1, \ldots , u_n$, which form a nonzero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields that are multiquadratic extensions of ${\mathbb{Q}}$ and improve and generalize a result of N. Rome (2017).

For an 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 0, Comment: 9 pages

A single exposure to statements is typically enough to increase their perceived truth. This Truth-by-Repetition (TBR) effect has long been assumed to occur only with statements whose truth value is unknown to participants. Contrary to this hypothesis, recent research found a TBR effect with statements known to be false. Of note, a recent model even posits that repetition could increase the perceived truth of highly implausible statements. As for now, however, no empirical evidence has reported a TBR effect for highly implausible statements. Here, we reasoned that one may be found provided a sensitive truth measure is used and statements are repeated more than just once. In a preregistered experiment, participants judged the truth of highly implausible statements on a 100-point scale, and these statements were either new to them or had been presented five times before the judgment task. We observed a TBR effect: truth judgments were higher for repeated statements than for new ones - even if all statements were still judged as false. Exploratory analyses additionally suggest that all participants were not equally prone to this TBR effect: about half the participants showed no or even a reverse effect. Overall, the results provide direct empirical evidence to the claim that repetition can increase perceived truth even for highly implausible statements, although not equally so for all participants and not to the point of making the statements look true.

We discover some properties of square numbers (perfect squares) if they are expressed in base 2s for s ≥ 3 and develop an algorithm that builds up the square root of a number (if it exists) from th...

Let n be a nonzero integer. A set of nonzero integers { a 1 , … , 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, we consider the question, for a given integer n which is not a perfect square, how large and how small can be the largest element in a D ( n ) -quadruple. We construct families of D ( n ) -quadruples in which the largest element is of order of magnitude | n | 3 , resp. | n | 2 ∕ 5 .

We have previously considered continued fractions with “numerator” a positive integer N, which we refer to as $${\text {cf}}_N$$ expansions. In particular, let E be a positive integer that is not a perfect square. For $$N > 1$$ , $$\sqrt{E}$$ has infinitely many $${\text {cf}}_N$$ expansions. There is a natural notion of the “best” $${\text {cf}}_N$$ expansion of $$\sqrt{E}$$ . We have conjectured, based on extensive numerical evidence, that such a best expansion is not always periodic. From this evidence, it is difficult to predict for which N this expansion will be periodic. We show here that for any such E, there are infinitely many values of N for which this expansion is indeed periodic, more precisely, periodic of period 1 or 2, and we obtain formulas for a subset of these expansions in terms of solutions to Pell’s equation $$x^2 - Ey^2 = 1$$ .

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.

In order to investigate multiplicative structures in additively large sets, Beiglbock et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded gaps (syndetic set) contains arbitrarily long geometric progressions. A result of Erdős implies that syndetic sets contain a 2-term geometric progression with integer common ratio, but we still do not know if they contain such a progression with common ratio being perfect square. In this article, we prove that for each $$k\in {\mathbb {N}}$$ , a syndetic set contains 2-term geometric progressions with common ratios of the form $$n^kr_1$$ and $$p^kr_2$$ , where $$p\in {\mathbb {P}}$$ (the set of primes), n is a composite number, $$r_1\equiv 1 \pmod {n}$$ , $$r_2\equiv 1\pmod {p}$$ , and $$r_1,r_2\in {\mathbb {N}}$$ . We also show that 2-syndetic sets (sets with bounded gap two) contain infinitely many 2-term geometric progressions with their respective common ratios being perfect squares.

This chapter deals with a remarkable property of the final digits of square numbers which could be described by the term “square number cycles”. Before these number cycles are discovered in the following, we will show some ideas for mental calculation of square numbers. The research on square number cycles can also be done with the help of pocket calculators or a spreadsheet, so if you are not interested in the tips for mental arithmetic, you can skip the first part of this article.

Consider $E$ a CM elliptic curve over $\mathbb{Q}$ . Assume that $\operatorname{rank}_{\mathbb{Q}}E\geq 1$ , and let $a\in E(\mathbb{Q})$ be a point of infinite order. For $p$ a rational prime, we denote by $\mathbb{F}_{p}$ the residue field at $p$ . If $E$ has good reduction at $p$ , let $\bar{E}$ be the reduction of $E$ at $p$ , let $\bar{a}$ be the reduction of $a(\operatorname{modulo}p$ ), and let $\langle \bar{a}\rangle $ be the subgroup of $\bar{E}(\mathbb{F}_{p})$ generated by $\bar{a}$ . Assume that $\mathbb{Q}(E[2])=\mathbb{Q}$ and $\mathbb{Q}(E[2],2^{-1}a)\neq \mathbb{Q}$ . Then in this article we obtain an asymptotic formula for the number of rational primes $p$ , with $p\leq x$ , for which ${\bar{E}(\mathbb{F}_{p})}/{\langle \bar{a}\rangle }$ is cyclic, and we prove that the number of primes $p$ , for which ${\bar{E}(\mathbb{F}_{p})}/{\langle \bar{a}\rangle }$ is cyclic, is infinite. This result is a generalization of the classical Artin’s primitive root conjecture, in the context of CM elliptic curves; that is, this result is an unconditional proof of Artin’s primitive root conjecture for CM elliptic curves. Artin’s conjecture states that, for any integer $a\neq \pm 1$ or a perfect square (or equivalently $a\neq \pm 1$ , and $\mathbb{Q}(\pm 1,\sqrt{a})=\mathbb{Q}(1[2],2^{-1}a)\neq \mathbb{Q}$ ), there are infinitely many primes $p$ for which $a$ is a primitive root (mod $p$ ), and an asymptotic formula for such primes is satisfied (this conjecture is not known for any specific $a$ ).

Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\dots,x\}$. Motivated by applications to factorisation algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should $N$ be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event, and conjectured that it in fact exhibits a sharp threshold in $N$. More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of $4/\pi + o(1)$ as $x \to \infty$, and made a conjecture regarding the location of the sharp threshold. In this paper we prove both of these conjectures, by determining the sharp threshold for making squares. Our proof combines techniques from combinatorics, probability and analytic number theory; in particular, we use the so-called method of self-correcting martingales in order to control the size of the 2-core of the random hypergraph that encodes the prime factors of our random numbers. Our method also gives a new (and completely different) proof of the upper bound in the main theorem of Croot, Granville, Pemantle and Tetali., Comment: 83 pages

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, 7 pages, revised version, to appear in Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM

Let d be a positive integer which is not a perfect square and n be any nonzero fixed integer. Then, the equation x 2 − d y 2 = n is known as the general Pell equation. In this paper, we give some criteria for class numbers of certain real quadratic fields to be greater than one, depending on the solvability of the general Pell equation, ideals in quadratic orders, and the period length of the simple continued fraction expansions of d .

For a nonzero integer $n$, a set of $m$ distinct nonzero integers $\{a_1,a_2,...,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1 \leq i < j \leq m$, is called a $D(n)$-$m$-tuple. In this paper, we show that there infinitely many essentially different quadruples which are simultaneously $D(n_1)$-quadruples and $D(n_2)$-quadruples with $n_1\neq n_2$., 8 pages

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 set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.

In this paper, we study products of consecutive integers and prove an optimal bound on their greatest common factors. In contrast, we obtain a modest upper bound for the greatest common factor in the perfect square situation. Using this and an upper bound on the size of solutions to hyperelliptic curves, we prove a gap principle when [Formula: see text] divides [Formula: see text] with some additional restrictions. We also obtain a stronger gap principle under the abc conjecture.

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.

In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs. Additionally, a degree condition providing the existence of an independent covering in the case where the graph has a square number of vertices is found., 10 pages, 6 figures

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices give rise to similar superalgebra structures, pointing to novel symmetry types in their complementary parts. In particular, this provides a unifying framework for the composite `most perfect square' symmetry and the related class of `reversible squares'; moreover, the semi-magic square algebra is identified as part of a 2-gradation of the general square matrix algebra. We derive explicit representation formulae for matrices of all symmetry types considered, which can be used to construct all such matrices.

In this paper, we provide some new expansions for the partial sums of Euler’s pentagonal number series and Gauss’ triangular and square numbers series. As applications, inequalities between certain partition functions are established.

Given a d × d quasiconvex quadratic form, d ≥ 3, we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero, then the form itself is an extremal quasiconvex quadratic form; i.e., it loses its quasiconvexity whenever a convex quadratic form is subtracted from it. In the special case d=3, we slightly weaken the condition; namely we prove that if the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square, then the form itself is an extremal quadratic form. In the case d=3 we also prove that if the determinant of the acoustic tensor of the form is identically zero, then the form is either extremal or polyconvex. Also, if the determinant of the acoustic tensor of the form is a perfect square, then the form is either extremal or polycovex, or is a sum of a rank-1 form and an extremal whose acoustic tensor determinant is identically zero. Here we use the notion of extremality introduced by Milton in 1990.© 2017 Wiley Periodicals, Inc.

Let n, a and b be positive integers. The pair (a; b) is called an integer partition of n into Diophantine pair if n = a+b, ab+1 is a perfect square and a > b. In this paper we give, for any positive integer n, a closed formula for the number of integer partitions into Diophantine pairs.Mathematics Subject Classication (2010): Primary 05A17; Secondary 11P83.Key words: Integer partitions, Diophantine pairs, τ (n), qD(n;m), qD(n; 2)

Two-dimensional version of the classical Mycielski theorem says that for every comeager or conull set $X\subseteq [0,1]^2$ there exists a perfect set $P\subseteq [0,1]$ such that $P\times P\subseteq X\cup \Delta$. We consider generalizations of this theorem by replacing a perfect square with a rectangle $A\times B$, where $A$ and $B$ are bodies of other types of trees with $A\subseteq B$. In particular, we show that for every comeager $G_\delta$ set $G\subseteq \omega^\omega\times \omega^\omega$ there exist a Miller tree $M$ and a uniformly perfect tree $P\subseteq M$ such that $[P]\times [M]\subseteq G\cup\Delta$ and that $P$ cannot be a Miller tree. In the case of measure we show that for every subset $F$ of $2^{\omega}\times 2^\omega$ of full measure there exists a uniformly perfect tree $P\subseteq 2^{

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, Mikic 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.

We find an infinite family of positive integers a such that concatenating a and a − 1 in base 10 (from left to right) results in a number that is a perfect square and estimates for such concatenations. F. L. was supported in part by grant CPRR160325161141 and an A-rated scientist award both from the NRF of South Africa and by grant no. 17-02804S of the Czech Granting Agency.