A Method to Solve the Diophantine Equation $ax^2-by^2+c=0$

Equivalence classes of solutions of the Diophantine equation $a^2+mb^2=c^2$ form an infinitely generated abelian group $G_m$, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-my^2=1$ generate a subgroup $P_m$ of $G_m$. We prove that $P_m$ and $G_m/P_m$ have infinite rank for all $m>1$. We also give several examples of $m$ for which $G_m/P_m$ has nontrivial torsion.

We discuss the history of attempts to solve the Pell equation using certain auxiliary equations that correspond, in modern terminology, to a second 2-descent.

In this paper we extend a result of Hirata-Kohno, Laishram, Shorey and Tijdeman on the Diophantine equation $n(n+d)...(n+(k-1)d)=by^2,$ where $n,d,k\geq 2$ and $y$ are positive integers such that $\gcd(n,d)=1.$

The main aim of this article is to find all solutions of the Diophantine equation $x^2 + p^k=y^n$ where $p \equiv 1 \pmod 4$, $\frac{p-1}{3}$ is a perfect square and the class number of $\mathbb{Z}[\sqrt{-p}]$ is $2$. In this article, I used a method involving prime factorization and class numbers which is different from using congruent number argument which is widely used in this type of problem.

One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling, based on previously developed machinery, Hilbert's question whether there exists a general algorithm for checking the solvability in integers of any diophantine equation. In this paper we describe techniques to prove the nonexistence of polynomials in two variables for some simple g...

This article deals with the history of certain aspects of the Pell equation X$^2$ - D y$^2$ = 4. We briefly discuss explicit units, and then study the history of Legendre's equations ax$^2$ - b y$^2$ = 4 with ab=D.

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a linear combination of $k$ quadratic forms of the aforementioned type, where $\lambda_i$'s are positive integers. The expressions for $a_k(n)$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ for different values of $k$ and $\lambda_...

Let $ \{L_n\}_{n\ge 0} $ be the sequence of Lucas numbers given by $ L_0=2, ~ L_1=1 $ and $ L_{n+2}=L_{n+1}+L_n $ for all $ n\ge 0 $. In the first paper, for an integer $d\geq 2$ which is square-free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^{2}-dy^{2}=\pm 1$ which is a product of two Lucas numbers, with a few exceptions that we completely characterize. Let $ \{P_m\}_{m\ge 0} $ be the sequence of Pell numbers gi...

In this study, we find continued fraction expansion of sqrt(d) when d=a^2+2a where a is positive integer. We consider the integer solutions of the Pell equation x^2-(a^2+2a)y^2=N when N={-1,+1,-4,+4}. We formulate the n-th solution (x_{n},y_{n}) by using the continued fraction expansion. We also formulate the n-th solution (x_{n},y_{n}) via the generalized Fibonacci and Lucas sequences.

In this document, we provide both the original German version of Ljunggren's article, "Ein Satz \"{u}ber die Diophantische Gleichung $Ax^{2}-By^{4}=C$ ($C=1,2,4$)", Tolfte Skand. Matematikerkongressen, Lund, 1953, pp.~188--194 (1954), as well as our English translation. It also contains a correction of his Satz 4.