Linked Questions

7
votes
2answers
2k views

Solve the Diophantine equation $ 3x^2 - 2y^2 =1 $

Solve $$ 3x^2 - 2y^2 =1 $$ in $ \mathbb{Z}$. How can we do it? ( All of answers gave me a great help. Thanks a lot kind stackexchangers.)
16
votes
3answers
243 views

Is there a simple proof that if $(b-a)(b+a) = ab - 1$, then $a, b$ must be Fibonacci numbers? [duplicate]

Consider the identity $(b-a)(b+a) = ab - 1$, where $a, b$ are nonnegative integers. We can also express this identity as $a^2 + ab - b^2 = 1$. This identity is clearly true when $a = F_{2i-1}$ and $...
2
votes
2answers
606 views

Solving the equation $ x^2-7y^2=-3 $ over integers

I'd like to solve the following Pell equation: $$ x^2-7y^2=-3 $$ Where $x$ and $y$ are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ...
4
votes
3answers
151 views

What are some books that are in the spirit of David A. Cox' “Primes of the Form $x^2+ny^2$”

David A. Cox "Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication." has a very good (at least to me, and many) methodology. He starts from page 1 asking a simple ...
4
votes
3answers
368 views

Algebraic proof of non-trivial solution to the Pell's equation

Let $d$ be a square-free positive integer, and consider the pairs $(x, y) \in \mathbb{Z}^2$ that satisfy: $$x^2 - dy^2 = 1$$ The existence of a non-trivial solution to this equation (i.e. distinct ...
3
votes
1answer
722 views

how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$

solve $ 3x^2+3xy-5y^2=55$ using number theory tools ,i have found the following $\Delta=3^2+4(5)(3)=9+60=69$ $d=69,u=1$ $w_{69}=\frac{1+\sqrt{69}}{2}$ $O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}...
5
votes
2answers
175 views

Classification of the positive integers not being the sum of four non-zero squares

It is well known that every positive integer is the sum of at most four perfect squares (including $1$). But which positive integers are not the sum of four non-zero perfect squares ($1$ is still ...
3
votes
1answer
312 views

General method for determining if $Ax^2 + Bx + C$ is square

Is there a general method for solving Diophantine equations in the form $Ax^2 + Bx + C = k^2$, preferably turning them into Pell's equations, when possible? For example, $2x^2 + x + 1 = k^2$ or $5x^2 +...
0
votes
2answers
348 views

Solutions to Diophantine Equations

I am looking for integer solutions to the equation $$x^2 = 5y^2 + 14y + 1$$ I know that Pell's Equation is of the form $x^2 - ny^2=1$ and that there exist algorithms to solve this equation. I was ...
2
votes
2answers
153 views

How to prove that the roots of this equation are integers?

Let there be an equation $a^2 + 4ab + b^2 - 121 = 0$ where I want to prove that a,b are integers. Then I want to find whether there are integer values of $b$ for which $a$ is also an integer. Let us ...
1
vote
1answer
165 views

Can I check whether integral solutions exist if I know a rational solution?

The pell-like equation $$x^2-101y^2=-71$$ has the rational solution $(x,y)=(\frac{25}{2},\frac{3}{2})$ Can I use this rational point to find out , whether an integral solution exists ? If yes, can ...
2
votes
2answers
91 views

Completeness proofs for the solutions of Diophantine Equations

In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations? For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set ...