Stancho Dimiev, Krassimir Markov
In this paper Gauss integers are used for to develop some geometry of full Erdoes graphs which vertics are points with integer coordinates. These graphs are called Diophantine figures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this note we find all the solutions of the Diophantine equation $x^4\pm y^4=iz^2$ using elliptic curves over $\mathbb Q(i)$. Also, using the same method we give a new proof of Hilbert's result that the equation $x^4\pm y^4=z^2$ has only trivial solutions in Gaussian integers.
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 studied in Franušić's previous paper, where it was shown that the extension to a Diophantine quadruple is unique (with an element $16k^3-4k$). The family of the triples of the same form $\{k-1, k+1, 16k^3-4k\}$ was already studied in rational integers.…
Solution of certain Diophantine equations in Gaussian integers
针对满足特定同余和勒让德符号条件的素数 p、q，论文利用两族椭圆曲线的挠点与 Q(i) 上的秩，证明两类四次高斯整数丢番图方程仅有平凡解。
Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_σ: y^2 = x (x+σp)(x+σq) . (σ=\pm 1). E=E_σis also denoted as E_+ or E_- when σ= +1or $-1.Here the Mordell-Weil group and the rank of the elliptic curve E over the Gauss field K=Q(\sqrt -1) (and over the rational field Q is determined in several cases; and results on solutions of related Diophantine equations and simultaneous Pellian equations will be given. The arithmetic constructs over Q of the elliptic curve E have been studied in the last paper1, the Selmer groups are determined, results on Mordell-Weil group,…
Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. Finally, we answer in the negative a question posed by Kleinbock, Lindenstrauss, and Weiss, by constructing a family …
Extravagance, irrationality and Diophantine approximation
该研究探讨高斯映射下测度与丢番图逼近的关系，证明部分商对数不可积的连分数混合测度下几乎所有数为刘维尔数，构造任意无理性指数的高斯不变遍历测度，并给出弱Renyi测度下的辛钦型二分法。
Our aim is to find a complex continued fraction algorithm finding all the best Diophantine approximations to a complex number. Using the sequence of minimal vectors in a two dimensional lattice over Gaussian integers, we obtain an algorithm defined on a submanifold of the space of unimodular two dimensional Gauss lattices. This submanifold is transverse to the diagonal flow. Thanks to the correspondence between minimal vectors and best Diophantine approximations, the algorithm finds all the best approximations to a complex number. A byproduct of the algorithm is the best constant for the compl…
We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.
Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-1 map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.
In this notes we make a comparison between the arithmetic properties of irrational numbers and their dynamical properties under the Gauss map. We show some equivalences between different classifications of irrational numbers such as the Diophantine classes and numbers admitting approximations by rational numbers at a given 'speed'. We also show that irrational numbers with finite upper Lyapunov exponent for the Gauss map satisfy a Diophantine condition.
Gauss-Dickson Codes
本文利用 Gauss 和 Dickson 丢番图系统的解，在有限域上构造出 1 检错 [2,1,2] 码和 1 纠错 [4,2,3] 码。
An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\mathbb Z[i]$
本文通过 AI 证明，在高斯整数环上，不存在算法判定 18 个未知数的多项式方程是否有解，将 Matiyasevich 和 Sun 的 20 变量定理改进为 18 变量。
A quadratic polynomial $Φ_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation $Φ_{a,b,c}(x,y,z)=n$ has an integer solution $x,y,z$ for any non negative integer $n$. In this article, we show that if $(a,b,c)=(2,2,6), (2,3,5)$ or $(2,3,7)$, then $Φ_{a,b,c}( x,y,z)$ is universal. These were conjectured by Sun in \cite {Sun}.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。