A Direct Method To Generate Pythagorean Triples And Its Generalization To Pythagorean Quadruples And n-tuples

Doubly regular Diophantine quadruples

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 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine m-tuples and certain family of elliptic curves, we show that there are infinitely many essentially different sets consisting of perfect squares which are simultaneously D(n_1)-quadruples and D(n_2)-quadruples with distinct non-zero squares n_1 and n_2.

Growth polynomials for additive quadruples and $(h,k)$-tuples

Consider the interval of integers $I_{m,n} = \{m, m+1, m+2,\ldots, m+n-1 \}$. For fixed integers $h,k,m$, and $c$, let $Φ_{h,k,m}^{(c)}(n)$ denote the number of solutions of the equation $(a_1+\cdots + a_h)- (a_{h+1} + \cdots + a_{h+k})=c$ with $a_i \in I_{m,n}$ for all $i=1,\ldots, h+k$. This is a polynomial in $n$ for all sufficiently large $n$, and the growth polynomial is constructed explicitly.

数域上的广义勾股三元组研究

On generalised Pythagorean triples over number fields

该研究将广义勾股三元组推广至数域场景，梳理拓展有理数域上的相关结果，实现数域下方程$E_{a,b,c}$的可解性判定、解的参数化与极小解求解，并辅以示例降低理解门槛。

Clifford Algebras and Euclid's Parameterization of Pythagorean Triples

We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra $\mathbb{R}_{2,1}$, whose minimal version may be conceptualized as a 4-dimensional real algebra of "kwaternions." We observe that this makes Euclid's parameterization the earliest appearance of the concept of spinors. We present an analogue of the "magic correspondence" for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unex…

有理数D(q)-四元组

Rational $D(q)$-quadruples

本文研究了有理数D(q)-四元组的性质，通过椭圆曲线上的点来描述满足特定乘积条件的四元组结构。

Erdos-Turan with a moving target, equidistribution of roots of reducible quadratics, and Diophantine quadruples

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also…

Extension of a Diophantine triple with the property $D(4)$

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

$\mathbb{Z}[\sqrt{4k + 2}]$ 中的 Diophantine $D(n)$-四元组

Diophantine $D(n)$-quadruples in $\mathbb{Z}[\sqrt{4k + 2}]$

本文研究了二次整数环 $\mathbb{Z}[\sqrt{d}]$ 中的 Diophantine $D(n)$-四元组，在特定条件下证明了对于满足特定同余关系的 $n$，存在无穷多个 $D(n)$-四元组，并给出了 $d=10$ 时的实例。