# Questions tagged [pell-type-equations]

Questions on finding integer solutions to bivariate equations of the form $x^2-Dy^2=a$.

243 questions
0answers
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### Books on polynomial Pell equations

I am interested in polynomial Pell equations and their links with polynomial continued fractions, Padè approximants and the conditions under which they are solvable. Could you suggest me some books ...
2answers
47 views

### Is every integer a solution to the generalized pell-like equation

Is every integer a solution to a generalized Pell-like equation, like, can we find integer solutions to $ax^2-by^2=n$, $a,b,x,y\in\mathbb{N}$ for any integer $n$? Specifically, it is difficult to ...
1answer
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### Trouble understanding generalised form of Pell’s equation

A Pell’s equation is a diophantine equation in $x$ and $y$ of the form $$x^2-d\cdot y^2=1$$ with $d$ a square-free integer. The fundamental solution of a Pell’s equation is the smallest (with ...
1answer
34 views

### An inequality leading into Pell's Equation

I need to show that for $\alpha$ irrational there are infinite coprime pairs $x,y\in\mathbb{Z}$, we have that $$\vert{\alpha-\frac{x}y}\vert\leqslant\frac{1}{y^2}$$ Hints are also appreciated
0answers
43 views

### Where is there a good set of questions on Pell's and nonlinear Diophantine equations?

Are there any resources of questions on the topic of Pell's and nonlinear Diophantine equations? I am looking for interesting results which are required to be solved. This is for a typical second ...
0answers
167 views

### How does this elementary method of solving Pell's equation compare with the classical methods?

The equation $x^2-dy^2=1$ will be transformed into a quadratic equation with the use of triangular numbers. The motivation to use the triangular numbers to express the squares $x^2$ and $y^2$ comes ...
3answers
348 views

### Demonstrate that these two Pell's equations have no integer solutions

I would like to demonstrate that the following four Pell's equations have no integer solutions: $x^2-82y^2=\pm2$ $x^2-82y^2=\pm3$ I do realise that such problems are often solved by algebraic ...
2answers
89 views

### How to solve pell type equation

Example $x^2-7y^2=2$ Find $x,y$ in general when $x,y$ are integer I don't know how to solve it , I need the method to help to solve it
1answer
93 views

### $x^2 + 3xy + y^2 = n$ Diophantine Equation

I was wondering if someone could direct me towards information regarding the $x^2 + 3xy +y^2 = n$ diophantine equation. Additionally, is there anything about the general case of these diophantine ...
0answers
32 views

1answer
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### Prove $(1+\sqrt{2})^k= \sqrt{n}+\sqrt{n-1}$ [duplicate]

Prove that for every $k \in \mathbb{N}$ there exists $n\in \mathbb{N}$ such that $(1+\sqrt{2})^k= \sqrt{n}+\sqrt{n-1}$ I tried to prove it using induction, but I could not move to the next step after ...
1answer
52 views

### A property of the negative pell equation

I want to show that if $(x,y)$ is a solution to the negative pell equation ($x^2-dy^2=-1)$, then $\frac{x}{y}$ is a convergent of the continued fraction expansion of $\sqrt{d}.$
3answers
104 views

### Triangle numbers that are squares of triangle numbers.

What are the triangle numbers the are squares of other triangle numbers? I have found $1^2=1$ and $6^2=36$, but other than these examples I can't find any other triangle numbers that are squares of ...
3answers
125 views

### Recurrence relation for Pell's equation $x^2-2y^2=1$

I am wondering how to find the recurrence relation for solutions for $x$ in the Pell's equation $x^2-2y^2=1$. I know the formula for the general term. It is $$\frac{(3+2\sqrt2)^n+(3-2\sqrt2)^n}{2}$$ ...
2answers
259 views

2answers
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### When is there a solution to the generalized Pell's equation?

Let's assume that $d>1$ is a squarefree integer. If I am given an integer $m$, is there a way to use algebraic number theory to determine whether or not $x^2-dy^2=m$ has a solution in integers? For ...
4answers
221 views

### How to show that $x^2 - 37y^2 =2$ does not have integer solutions

We need to prove that $x^2 - 37y^2 =2$ does not have integer solutions. I have two angles I thought about approaching it from: Since 37 is prime, I can show that for $x$ not divisible by $37$, we ...
0answers
63 views

### A question about principality of ideals dividing $(p)$ in imaginary quadratic field

Exercise 8 of chapter IX.3 of Cohn's book Advanced Number Theory reads: Show that if the fundamental discriminant $d = g_1 g_2 < 0$ satisfies $(g_1/p) = (g_2/p) = -1$ for a prime $p$, then the ...
1answer
291 views

### Existence of Solution to Generalized Pell's Equation

Consider the generalized Pell's equation $x^2 - dy^2 = n$ for $d \in \mathbb{Z}_{> 0}$ and $n \in \mathbb{Z} \setminus \{0\}$. When does this equation have a solution for $x,y$ over $\mathbb{Z}$? ...
2answers
93 views

### Fundamental solution to specific Pell equation

I want to find the fundamental solution of: $$x^2 - dy^2 = 1$$ where $d$ is of the form $d = m^2 + 2$. I know how to solve these kind of problems using the continued fraction of $\sqrt{d}$, but ...
1answer
44 views

### Show that $x^2-dy^2 = -2$ with $d = m^2+2$ has infinitetly many integer solutions

I want to show that for $d= m^2+2$ the equation $$x^2-dy^2 = -2$$ has infinitetly many integer solutions. By trying out one can see that $x=\pm m, y=\pm 1$ are solutions, but how do we know that ...
1answer
434 views