Ramanujan-Nagell type equations and perfect numbers

On the solutions of certain Lebesgue-Ramanujan-Nagell equations

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.

广义Lebesgue-Ramanujan-Nagell方程的完全解

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。

A modular approach to the generalized Ramanujan-Nagell equation

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.

On the number of solutions of the generalized Ramanujan-Nagell equation $D_{1}x^2+D_{2}^{m}=2^{n+2}$}

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 $N(13,3)=N(31,97)=3$.

一类 Lebesgue-Ramanujan-Nagell 型方程的完整解

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。

Some experiments with Ramanujan-Nagell type Diophantine equations

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…

Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

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…

关于一类 Lebesgue-Ramanujan-Nagell 方程

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。

Ramanujan–Nagell 定理在 Lean 4 中的形式化证明

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。

Differences between perfect powers : the Lebesgue-Nagell Equation

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 $11$.

A robust implementation for solving the $S$-unit equation and several applications

Let $K$ be a number field, and $S$ a finite set of places in $K$ containing all infinite places. We present an implementation for solving the $S$-unit equation $x + y = 1$, $x,y \in\mathscr{O}_{K,S}^\times$ in the computer algebra package SageMath. This paper outlines the mathematical basis for the implementation. We discuss and reference the results of extensive computations, including exponent bounds for solutions in many fields of small degree for small sets $S$. As an application, we prove an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded disc…

关于三个连续幂数的初等注记

An elementary note on three consecutive powerful numbers

本文利用初等方法及相关数论结果，推广了T. H. Chan关于三个连续幂数的结论，讨论了若干丢番图方程的解。

Almost-Ramanujan Graphs and Prime Gaps

The method of Murty and Cioabă shows how one can use results about gaps between primes to construct families of almost-Ramanujan graphs. In this paper we give a simpler construction which avoids the search for perfect matchings and thus eliminates the need for computation. A couple of recent explicit bounds on the gap between consecutive primes are then used to give the construction of $k$-regular families with explicit lower bounds on the spectral gaps. We then show that a result of Ben-Aroya and Ta-Shma can be improved using our simpler construction on the assumption of the Riemann Hypothesi…

非素数幂字母表上的完美码：一种基于丢番图方程的方法

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。