Massimo Salvi
In this paper we show a way to generalize the linear Diophantine equation a1x1+a2x2+...+anxn=d . We deal with the nonlinear Diophantine equation det|A X|=+-d , which generalizes the linear one, and we give a necessary and sufficient condition for its solvabiliy. We show how the linear equation can be considered as a particular case of the nonlinear equation.
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We consider Diophantine quintuples $\{a, b, c, d, e\}$, sets of distinct positive integers the product of any two elements of which is one less than a perfect square. Triples of the first kind are the subsets $\{a, b, d\}$ with $d> b^{5}$. We show that there are no Diophantine quintuples containing triples of the first kind.
On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers
本文完整求解了由三个连续$k$--广义卢卡斯数幂构成的非线性丢番图方程，给出了所有非负整数解。
This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood. Author found the upper bound of the number of natural solutions of inhomogeneous algebraic Diophantine diagonal equations with explicit variable. He developed a method in the paper, which allows you to perform the low estimate of the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. Author obtained a lower estim…
We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarnik type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess's bound on short character sums.
We solve a family of quadratic Diophantine equations associated to a simple kind of games. We show that the ternary case, in many ways, is the most interesting and the least arbitrary member of the family.
The paper deals with best one--sided (lower or upper) Diophantine approximations of the $\ell$-th kind ($\ell\in\mathbb{N}$). We use the ordinary continued fraction expansions to formulate explicit criteria for a fraction $\frac{p}{q}\in\mathbb{Q}$ to be a best lower or upper Diophantine approximation of the $\ell$-th kind to a given $α\in\mathbb{R}$. The sets of best lower and upper approximations are examined in terms of their cardinalities and metric properties. Applying our results in spectral analysis, we obtain an explanation for the rarity of so-called Bethe--Sommerfeld quantum graphs.
We prove general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters betaN, grounding on combinatorial properties of positive density sets and IP sets. Necessary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers *N.
Singmaster-type results for Stirling numbers and some related diophantine equations
本文受Singmaster启发，研究整数在两类Stirling数中的出现次数，给出上界并提出相关问题，同时收集相关丢番图方程的数值结果与猜想。
A perfect cuboid is a rectangular parallelepiped. Its edges, its face diagonals, and its space diagonal are of integer lengths. None of such cuboids is known thus far, though the system of Diophantine equations describing them is easily written. The cuboid characteristic equation is a twelfth degree Diophantine equation derived from the initial cuboid equations and equivalent to them. In the case of the second cuboid conjecture it reduces to a tenth degree equation. This equation comprises two parameters. Previously various asymptotics for roots of this equation were studied as its parameters …
On Pillai's Problem involving Lucas sequences of the second kind
本文研究第二类Lucas序列与完全幂之差的Pillai型丢番图方程，证明在至少存在三个解时，解的大小和特征多项式系数均有上界。
Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado's characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado's theorem. We verify that such a conjecture hold for the equations $x^{2}-xy+ax+by+cz=0$ and $x^{2}-y^{2}+ax+by+cz=0$ for $a,b,c\in…
Diophantine estimates on shifts of trigonometric polynomials on $\mathbb{T}^d$
本文在环面上建立三角多项式平移及其平方根的丢番图估计，源于拟周期算子谱分析，可推广Bourgain结果至非线性方程。
Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Padé approximations of the second kind.
The Hausdorff dimension of an exceptional set of periods for which convergence of a formal solution to an inhomogeneous wave equation in n spatial and one temporal dimension is problematic, is determined along with conditions which the periods must satisfy to ensure the solvability of the inhomogeneous wave equation by a smooth periodic function. To derive this information, a complete metric theory for a related fully nonlinear Diophantine approximation problem involving perfect squares is established.
We consider Diophantine equations of the kind $|F(x,y)|= m$, where $F(X,Y )\in \bz [X,Y]$ is a homogeneous polynomial of degree $d\ge 3$ that has non-zero discriminant and $m$ is a positive integer. We prove results that simplify those of Stewart and provide heuristics for a conjecture of Stewart.
Core abaci and Diophantine equations I: fundamental weight
本文定义并研究了任意电荷下的经典仿射类型核心abaci，通过关联核心abaci与权重及仿射威利群元素，证明了β的高度等于原子长度，并解决了广义开放问题，同时建立了经典仿射类型的丢番图方程，并证明某些类别的解由核心abaci完全参数化。
We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the generating function for those numbers. We obtain explicit expressions for the coefficients appearing in those differential equations and the degenerate Bernoulli numbers of the second kind. In addition, as an application and from those differential equations we have an identity expressing the degenerate Bernoulli numbers of the second kind in terms of those numb…
We consider the set of Diophantine equations that arise in the context of the barotropic vorticity equation on periodic domains, when nonlinear wave interactions are studied to leading order in the amplitudes. The solutions to this set of Diophantine equations are of interest in atmosphere (Rossby waves) and Tokamak plasmas (drift waves), because they provide the values of the spectral wavevectors that interact resonantly via three-wave interactions. These come in "triads", i.e., groups of three wavevectors. We provide the full solution to the Diophantine equations in the case of infinite Ross…
We prove an ultrametric q-difference version of the Maillet-Malgrange theorem, on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since °_q and \ord_q define two valuations on {\mathbb C}(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the f…
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