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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

To clarify the effect of the N deficit on the amount of square Bt insecticidal protein, different N application rates (0, 75, 150, 225, and 300 kg ha–1) were imposed on the conventional cultivar Sikang 1 (SK-1) and hybrid cultivar Sikang 3 (SK-3) during 2015–2016 cotton growth seasons. Under different N application rates, the square number per plant, square volume and square dry weight reduced when the N rates decreased from conventional rate (300 kg ha–1) to 0 kg ha–1. And the square Bt protein content decreased accordingly. The analysis of N metabolism showed that soluble protein content, GPT and GOT activities decreased, free amino acid, peptidase and protease activities increased under N deficit. Correlation analysis indicated that the reduced Bt protein content under N deficit was related to altered N metabolism. In conclusion, square development and the amount of square Bt toxin both decreased under N deficit, indicating that promoting the square development under appropriate N application rate would also promote the insect resistance during squaring stage.

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.

The τ5 parameter, first proposed by Addison and coworkers, is the principal measure of the geometry of five-coordinate transition metal complexes, with τ5 = 0 said to describe a perfect square pyramidal geometry and τ5 = 1 a perfect trigonal pyramidal geometry. Therefore, the geometries of all five-coordinate complexes are assumed to lie on a continuum between these two extremes. Herein we show that there are a significant number of examples of transition metal complexes having τ5 > 1, leading to an equatorially distorted trigonal bipyramidal geometry with the transition metal ion lying out of the plane of the equatorial donor atoms. We also show that complexes having τ5 = 0 and displaying perfect square pyramidal geometry are very much the exception, and that the majority of complexes for which τ5 = 0 have the metal ion sitting above the mean plane of the donor atoms in the square plane, in a basally distorted square pyramidal geometry. Density functional theory computations on a number of these complexes show that the structural distortions are inherent features of the complexes, and not merely the result of intermolecular interactions.

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

Let G = (V, E) be a simple connected graph with p vertices and q edges. If G1, G2, …., Gn are connected edge disjoint subgraphs of G with E(G) = E(G1) \(\cup\)E(G2)\(\cup\)….\(\cup\)E(Gn), then (G1, G2, ……, Gn) is said to be a decomposition of G. A decomposition (G1, G2, ……, Gn) of G is said to be continuous monotonic decomposition if each Gi is connected and |E(Gi)| = i for every i = 1, 2, …., n. [1]. In this chapter, we introduced the concept Arithmetic odd star decomposition. A decomposition (G1, G2, ……, Gn) of G is said be an Arithmetic decomposition if |E(Gi) = a + (i-1)d, for every I = 1, 2 ……, n and a, d \(\in\) Z. Clearly q=\(\frac{n}{2}\)[2a + (n - 1) d]. If a = 1 and d = 1, then q = \(\frac{n(n+1)}{2}\). In this chapter, we study the graphs with a = 1 and d = 2. Then q = n2. That is, the number of edges of G is a perfect square.

The public parameters of the RSA cryptosystem are represented by the pair of integers N and e . In this work, first we show that if e satisfies the Diophantine equation of the form e x 2 − ϕ ( N ) y 2 = z for appropriate values of x , y and z under certain specified conditions, then one is able to factor N . That is, the unknown y x can be found amongst the convergents of e N via continued fractions algorithm. Consequently, Coppersmith’s theorem is applied to solve for prime factors p and q in polynomial time. We also report a second weakness that enabled us to factor k instances of RSA moduli simultaneously from the given ( N i , e i ) for i = 1 , 2 , ⋯ , k and a fixed x that fulfills the Diophantine equation e i x 2 − y i 2 ϕ ( N i ) = z i . This weakness was identified by solving the simultaneous Diophantine approximations using the lattice basis reduction technique. We note that this work extends the bound of insecure RSA decryption exponents.

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.

Dicuil was a ninth-century Irish monk who taught at the Carolingian school of Louis the Pious. He wrote a Computus or astronomical treatise in Latin in about 814–16, which contains a chapter on tri...

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

The Rivest-Shamir-Adleman (RSA) cryptosystem is one of the strong encryption approaches currently being used for secure data transmission over an insecure channel. The difficulty encountered in breaking RSA derives from the difficulty in finding a polynomial time for integer factorization. In integer factorization for RSA, given an odd composite number n, the goal is to find two prime numbers p and q such that n = p q. In this paper, we study several integer factorization algorithms that are based on Fermat’s strategy, and do the following: First, we classify these algorithms into three groups: Fermat, Fermat with sieving, and Fermat without perfect square. Second, we conduct extensive experimental studies on nine different integer factorization algorithms and measure the performance of each algorithm based on two parameters: the number of bits for the odd composite number n, and the number of bits for the difference between two prime factors, p and q. The results obtained by the algorithms when applied to five different data sets for each factor reveal that the algorithm that showed the best performance is the algorithms based on (1) the sieving of odd and even numbers strategy, and (2) Euler’s theorem with percentage of improvement of 44% and 36%, respectively compared to the original Fermat factorization algorithm. Finally, the future directions of research and development are presented.

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.

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like the square root of 2, is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers – specifically, the radicands of radical expressions – as natural numbers. Strategy self-reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task-specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.

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.

This study explores the grasp of square roots among eleven students in a remedial mathematics course, with a special focus on where the same student generated apparently conflicting responses. Building on the commognitive framework, the analysis distinguished between routines that individual students consistently implemented in situations where roots “stood alone” and where they were incorporated in more compound exercises, where roots were extracted from square numbers and from squared radicands, where roots were applied to monomials and binomials, and where parameters named with different letters were involved. Differences were found in routines’ degree of objectification, procedures, and tasks. These differences are explained with a theoretical account, suggesting that what may seem as a conflict within a student’s discourse could be a sensible difference of actions taken in situations that this student construed as different. The contribution of this study to the body of knowledge on teaching and learning of roots and to commognitive research is discussed.