Yann Bugeaud, Maurice Mignotte, Samir Siksek
We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.
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
The generalized Lebesgue-Ramanujan-Nagell equation is an important type of polynomial-exponential Diophantine equation in number theory. In this survey, the recent results and some unsolved problems of this equation are given.
We find all positive integer solutions in $x, y$ and $n$ of $x^2+19^{2k+1}=4y^{n}$ for any non-negative integer $k$.
In this paper, we study the generalized Lebesgue-Nagell equation \[ x^2+7^{2k+1}=y^n. \] This is the last case of equations of the form $x^2+q^{2k+1}=y^n$ with $k\geq0$ and $q>0$ where $\mathbb{Q}(\sqrt{-q})$ has class number one. Our proof is based on the modular method and the symplectic argument.
It is well-known that for $p=1, 2, 3, 7, 11, 19, 43, 67, 163$, the class number of $\mathbb{Q}(\sqrt{-p})$ is one. We use this fact to determine all the solutions of $x^2+p^m=4y^n$ in non-negative integers $x, y, m$ and $n$.
We describe a computationally efficient approach to resolving equations of the form $C_1x^2 + C_2 = y^n$ in coprime integers, for fixed values of $C_1$, $C_2$ subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.
We completely solve the Diophantine equation $x^2+2^k11^\ell19^m=y^n$ in integers $x,y\geq 1;~ k,\ell, m\geq 0~$ and $n\geq 3$ with $\gcd(x,y)=1$, except the case $2\mid k, 2\nmid \ell m$ and $5\mid n$. We use this result to recover some earlier results in the same direction.
Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type $x^2=Ak^{n}+B$ with many solutions. Here, $A,B\in\Z$ (thus $A, B$ are not necessarily positive) and $k\in\Z_{\geq 2}$ are given integers. In particular, we prove that for each $k$ there exists an infinite set $\cal{S}$ containing pairs of integers $(A, B)$ such that for each $(A,B)\in \cal{S}$ we have $\gcd(A,B)$ is square-free and the Diophantine equation $x^2=Ak^n+B$ h…
On the complete solutions of a generalized Lebesgue-Ramanujan-Nagell equation
本文研究广义Lebesgue-Ramanujan-Nagell方程，通过结合经典本原因子定理与计算机搜索，完整求解了该方程及其他相关类型方程的所有整数解。
We develop a variety of new techniques to treat Diophantine equations of the shape $x^2+D =y^n$, based upon bounds for linear forms in $p$-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers $x$ and $y$, and $n \geq 3$, with the property that $y^n > x^2$ and $x^2-y^n$ has no prime divisor exceeding $11$.
Complete solutions of a Lebesgue-Ramanujan-Nagell type equation
本文完整求解了一类含素数 5、13、17 的 Lebesgue-Ramanujan-Nagell 型指数丢番图方程，借助 Lehmer 序列本原除子理论与椭圆曲线 S-整点方法给出全部整数解。
On the Lebesgue-Nagell equation $x^2-2 = y^p$
本文研究 Lebesgue-Nagell 方程 $x^2-2=y^p$，通过下界方法确认了平凡解猜想在 $p\leq 13$ 和 $p>911$ 时成立，并证明非平凡解满足 $y>10^{1000}$。
We shall show that, for any positive integer $D>0$ and any primes $p_1, p_2$ not dividing $D$, the diophantine equation $x^2+D=2^s p_1^k p_2^l$ has at most $63$ integer solutions $(x, k, l, s)$ with $x, k, l\geq 0$ and $s\in \{0, 2\}$.
Interpolation determinants and the Lebesgue--Nagell equation $a^2-D=b^p$
本文证明奇素数$p$下$a^2-2=b^p$仅有平凡解，并解决$D=3,5$的类似方程，提出一般方法，改进插值行列式技术。
Given odd, coprime integers $a$, $b$ ($a>0$), we consider the Diophantine equation $ax^2+b^{2l}=4y^n$, $x, y\in\Bbb Z$, $l \in \Bbb N$, $n$ odd prime, $\gcd(x,y)=1$. We completely solve the above Diophantine equation for $a\in\{7,11,19,43,67,163\}$, and $b$ a power of an odd prime, under the conditions $2^{n-1}b^l\not\equiv \pm 1(\mod a)$ and $\gcd(n,b)=1$. For other square-free integers $a>3$ and $b$ a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers $x$, $y$ with ($\gcd(x,y)=1$), $l\in\mathbb{N}$ and all odd primes $n>3$, satisfying $2^{n-…
$\mathbb{Q}$-curves and the Lebesgue-Nagell equation
本文通过构造实二次域上的Frey-Hellegouarch $\mathbb{Q}$-曲线并运用多Frey模方法，完整求解了$q=41$和$q=97$时Lebesgue-Nagell方程$x^2-q^{2k+1}=y^n$的所有整数解。
The Diophantine equation (x^n-1)/(x-1)=y^q has four known solutions in integers x, y, q and n with |x|, |y|, q > 1 and n > 2. Whilst we expect that there are, in fact, no more solutions, such a result is well beyond current technology. In this paper, we prove that if (x,y,n,q) is a solution to this equation, then n has three or fewer prime divisors, counted with multiplicity. This improves a result of Bugeaud and Mihailescu.
On a class of Lebesgue-Ramanujan-Nagell equations
本文研究一类 Lebesgue-Ramanujan-Nagell 方程，利用二次域类数条件给出多种情形下的完整求解，并讨论特定参数下的可解性。
In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we u…
A formal proof of the Ramanujan--Nagell theorem in Lean 4
利用 Lean 交互式定理证明器和 Mathlib 库，完整形式化了 Ramanujan–Nagell 定理，即丢番图方程 $x^2 + 7 = 2^n$ 仅有五个整数解。
On the Ternary Purely Exponential Diophantine Equation $(ak)^x+(bk)^y=((a+b)k)^z$ with Prime Powers $a$ and $b$
该研究结合初等数论方法与经典丢番图方程结果、Lucas序列新性质，证明了特定素数幂条件下一类三元纯指数丢番图方程仅有正整数解$(1,1,1)$，部分验证了已有猜想。
In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.
Hausdorff measure and Fourier dimensions of limsup sets arising in weighted and multiplicative Diophantine approximation
本文研究了加权和乘法Diophantine逼近中出现的limsup集的Lebesgue测度、Hausdorff测度和Fourier维度，提供了零-全律判定准则，并确定了这些集的精确Fourier维度。
We consider the Diophantine equation X^n - 1 = B.Z^n, where B in Z is understood as a parameter. We prove that if the equation has a solution, then either the Euler totient of the radical, phi(rad (B)), has a common divisor with the exponent n, or the exponent is a prime and the solution stems from a solution to the diagonal case of the Nagell-Ljunggren equation: (X^n-1)/(X-1) = n^e.Y^n; e = 0 or 1. This allows us to apply recent results on this equation to the binary Thue equation in question. In particular, we can then display parametrized families for which the Thue equation has no solution…
Weighted Fruit Diophantine Equations and Hyperelliptic Curves
研究加权水果丢番图方程在特定参数条件下的整数解不存在性，通过关联超椭圆曲线将解的不存在性转化为有理扭点的结果。
Under the assumption that the approximating function $ψ$ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of the set of $ψ$-approximable matrices in $\R^{mn}$. The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on $ψ$ is absolutely necessary when $m=n=1$. On the other hand, it is known that monotonicity is not necessary when $n > 2$ (Sprindzuk) or when $n=1$ and $m>1$ (Gallagher). Surprisingly, when $n=2$ the situation is unresolved. We deal with this remaining case and thereby remove all …
This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(α)$, where $S(α)$ is the set of all integers of the form $\lfloor αn^2 \rfloor$ for $n\geq α^{-1/2}$ and $\lfloor x\rfloor$ denotes the integer part of $x$. We show that $T(α)$, the set of all such tuples, is infinite for all $α\in (0,1)\cap \mathbb{Q}$ and for almost all $α\in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $α>0$ such that $T(α)$ is finite, then there is no perfect Euler brick. W…
We present shrinking targets results for general systems with the emphasis on applications for IETs (interval exchange transformations) $(J,T)$, $J=[0,1)$. In particular, we prove that if an IET $(J,T)$ is ergodic (relative to the Lebesgue measure $\lam$), then the equality \[ \liminf_{n\to\infty}\limits n |T^n(x)-y|=0 \tag{A1} \] holds for $\lam\ttimes\lam$-a. a. $(x,y)\in J^2$. The ergodicity assumption is essential: the result does not extend to all minimal IETs. The factor $n$ in (A1) is optimal (e. g., it cannot be replaced by $n \ln(\ln(\ln n))$. On the other hand, for Lebesgue almost al…
Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(ψ_n)_{n \in \mathbb{N}}$ and differentiable functions $(φ_n: J \to \mathbb{R})_{n \in \mathbb{N}}$ so that for Lebesgue-a.e. $θ\in J$, the inequality $\| nθ+ φ_n(θ) \| \leq ψ_n$ has infinitely many solutions. The main novelty is that the magnitude of the perturbation $|φ_n(θ)|$ is allowed to exceed $ψ_n$, changing the usual "shrinking targets" problem into a "shifting targets" problem. As an application of the main resul…
A uniform metrical theorem in multiplicative Diophantine approximation
本文研究了勒贝格通用二维实数对的乘性丢番图逼近问题，给出了满足特定增长条件下相关小值乘积数量的渐近公式。
Perfect codes over non-prime power alphabets: an approach based on Diophantine equations
本文利用广义拉马努金-纳格尔方程和计算数论方法，证明172个新非素数幂字母表上不存在完美2错误纠正码，并证明任意固定字母表大小上此类码至多有限个。
On equations $(-1)^αp^x+(-1)^β(2^k(2p+1))^y=z^2$ with Sophie Germain prime $p$
本文研究含索菲·热尔曼素数的一类丢番图方程，求解了p=2时的方程及奇素数下四类方程的所有非负整数解，验证了对应素数的存在性并提出相关无穷性猜想。
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。