相关论文: A Generalised Stern-Brocot Tree from Regular Dioph…
In this note we present a construction of an infinite family of diagonal quintic threefolds defined over $\Q$ each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples $B=(B_{0},…
For fixed integers $D \geq 0$ and $c \geq 3$, we demonstrate how to use $2$-adic valuation trees of sequences to analyze Diophantine equations of the form $x^2+D=2^cy$ and $x^3+D=2^cy$, for $y$ odd. Further, we show for what values $D \in…
For $D$ a natural number that is not a perfect square and for $k$ a non-zero integer, consider the subset $\mathbb{Z}_k(\sqrt{D})$ of the quadratic integer ring $\mathbb{Z}(\sqrt{D})$ consisting of elements $x+y\sqrt{D}$ for which $x^2 -…
In this paper we show that, for any fixed $1<c<967/805$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\varepsilon \end{equation*}…
Recently, Hajdu and S\'{a}rk\"{o}zy studied the multiplicative decompositions of polynomial sequences. In particular, they showed that when $k \geq 3$, each infinite subset of $\{x^k+1: x \in \mathbb{N}\}$ is multiplicatively irreducible.…
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…
Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial…
For a prime p, a Diophantine m-tuple in $\mathbb{F}_p$ is a set of m nonzero elements of $\mathbb{F}_p$ with the property that the product of any two of its distinct elements is one less than a square. In this paper, we present formulas for…
Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient…
We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…
In analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset $C$ of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of $C$, then it…
An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…
This paper is concerned with the diophantine system, $\sum_{i=1}^{s_1} x_i^r=\sum_{i=1}^{s_2} y_i^r,\, r=1,\,2,\,\ldots,\,k, $ where $s_1$ and $s_2$ are integers such that the total number of terms on both sides, that is, $s_1+s_2,$ is as…
Given a lower-triangular matrix of real numbers, one can ask the following four total-positivity questions: total positivity of the triangle itself; total positivity of its row-reversal; Toeplitz-total positivity of its row sequences…
We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…
We revisit the problem of rational search: given an unknown rational number $\alpha = \frac{a}{b} \in (0,1)$ with $b \leq n$, the goal is to identify $\alpha$ using comparison queries of the form ``$\beta \leq \alpha$?''. The problem has…
There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…
Farey sequences, Stern-Brocot sequences, the Calkin-Wilf sequences are shown to be generated via almost identical second order recurrence relations. These sequences have combinatorial, computational, and geometric applications, and are…
The Diophantine equation 4/n=1/x+1/y+1/z for a Pythagorean prime n is split into two independent Diophantine equations, which correspond to two different types of solution. The solvability of these equations forces certain restrictions on…
In recreational mathematics, a normal magic square is an $n \times n$ square matrix whose entries are distinctly the integers $1 \ldots n^2$, such that each row, column, and major and minor traces sum to one constant $\mu$. It has been…