Maohua Le, Gökhan Soydan
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.
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We prove some results on the distance between two perfect numbers using Ramanujan-Nagell type equations.
The paper consists of two parts. The aim of the first, and main, part is to explain, in an elementary way, Hasse's proof of Ramanujan-Nagell's Theorem. In the second part, we formulate some natural extensions of Ramanujan-Nagell's equation.
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 $s\in \{0, 2\}$.
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.
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 $n$.
Let $k$ be a fixed positive integer with $k>1$. In this paper, using various elementary methods in number theory, we give criteria under which the equation $x^2+(2k-1)^y=k^z$ has no positive integer solutions $(x,y,z)$ with $y\in\{3,5\}$.
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.
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 find all positive integer solutions in $x, y$ and $n$ of $x^2+19^{2k+1}=4y^{n}$ for any non-negative integer $k$.
Let $k$ be a positive integer. In this paper, using the modular approach, we prove that if $k\equiv 0 \pmod{4}$, $30< k<724$ and $2k-1$ is an odd prime power, then under the GRH, the equation $x^2+(2k-1)^y=k^z$ has only one positive integer solution $(x,y,z)=(k-1,1,2)$. The above results solve some difficult cases of Terai's conecture concerning this equation.
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.
Let $D_{1}$, $D_{2}$ be coprime odd integers with min$(D_{1},D_{2})>1$, and let $N(D_{1},D_{2})$ denote the number of positive integer solutions $(x, m, n)$ of the equation $D_{1}x^2+D_{2}^{m}=2^{n+2}$. In this paper, we prove that $N(D_{1},D_{2})\leq 2$ except for $N(3,5)=N(5,3)=4$ and $N(13,3)=N(31,97)=3$.
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。
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 $11$.
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…
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…
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 $y>10^{1000}$。
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-…
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。
$\mathbb{Q}$-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。
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。
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