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Questions tagged [pell-type-equations]

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

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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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
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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 ...
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How do I prove that $\mathcal{P}_d(\mathbb{Z})^+\neq\varnothing$ and that Pell's equation solutions are convergents

I recently had to skip a number theory lecture because I was sick, and they proved that the set $$\mathcal{P}_d(\mathbb{Z})^+=\lbrace(x_0,y_0)\in\mathbb{Z}^2_{\geq1}:x_0^2-dy_0^2=1\rbrace\neq\...
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pell's equation converge

Explain Why pell's equation $x_n+ny_n=1$, $(x_n/y_n)^2$ is coverge to n as n increase For example, n=11 the answer $(x_n/y_n)^2$ is very close to 11 when n increase.
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Conditions on those Pell-type equations to admit solutions

I am facing those generalised Pell equations, $a^2-Db^2=-8$ and $x^2-Dy^2=8$, where $D=t^2+8t$ for an odd $t$, $t>2$. In particular, I would like to find the cases where both equations admit (...
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Evaluating $\liminf_{n\to\infty}n\{n\sqrt2\}$

How can we evaluate $$\liminf_{n\to\infty}n\{n\sqrt2\},$$where $\{\cdot\}$ denotes the fractional part of $\cdot$? The first thing came to my mind is Pell's equation $x^2-2y^2=1$. Knowing that $\...
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If $2n+1$ and $3n+1$ are perfect squares, then prove that $8|n$.

If for some number $n\in \mathbb N$, the numbers $2n+1$ and $3n+1$ are perfect squares of integers, then prove that $8|n$. if $2n+1=m^2$ and $3n+1=k^2$ then $k^2-m^2=3n-2n+1-1=n$ now I need to show ...
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How do i compute $f_n = 3f_{n-1} + 2\sqrt{2f_{n-1}^2 - 2}$ for $n$ around $10^{18}$?

So I have the recurrence $$f_n = \begin{cases} 3f_{n-1} + 2\sqrt{2f_{n-1}^2 - 2}, &n > 1\\ 3, &n = 1\\ \end{cases}$$ and I need to compute it in $O(\lg n)$, for $n$ as big as $10^{18}$. I ...
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Has my conjecture using Pell's Equation been discovered before

We have $$\sqrt{d} = \frac{x}{y} - \frac{1}{f_0\cdot y} - \frac{1}{f_0\cdot f_1\cdot y}- \ldots - \frac{1}{f_0\cdot f_1\cdot\ldots\cdot f_n\cdot y}-\ldots\,,$$ where $$f_0 = 2x\,,$$ $$f_{n+1} = (f_n)^...
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138 views

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 ...
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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 $\sqrt{d}.$
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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 ...
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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 $$\frac{(3+2\sqrt2)^n+(3-2\sqrt2)^n}{2}$$ ...
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Solve $a^2 - 2b^2 - 3 c^2 + 6 d^2 =1 $ over integers $a,b,c,d \in \mathbb{Z}$

Are we able to completely solve this variant of Pell equation? $$ x_1^2 - 2x_2^2 - 3x_3^2 + 6x_4^2 = 1 $$ This has an interpretation as is related to the fundamental unit equation of $\mathbb{Q}(\sqrt{...
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Pell's equation (or a special case of a second order diophantine equation)

Question Find integers $x,y$ such that $$x^2-119y^2=1.$$ So far I've tried computing the continued fraction of $\sqrt{119}$ to find the minimal solution, but either I messed up or I don't know where ...
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72 views

Understanding a particular method of solving generalized version of Pell's equation

So, I have understood how to solve Diophantine equations of the form $$x^2-Dy^2=1.$$ However, when I was reading the solution of the generalized Pell's equation $$x^2-Dy^2=c,$$ I got stuck. I knew how ...
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116 views

Diophantine equation $x^3+x+a^2=y^2$

Prob: Show that for any positive integer $a$, Diophantine equation $$x^3+x+a^2=y^2$$ has at least one solution $(x, y)$, where $x, y$ are positive integers. Source: My teacher. Attempt: First I ...
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Pell type equation about prime [duplicate]

Let $p=4k+1$ be a prime number such that $p=a^2+b^2$ , with $a$ an odd integer. Prove that the equation $$x^2-py^2=a$$ has at least a solution in $\mathbb{Z}$. Only a little progress (maybe useless): ...
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About the diophantine $x^2 + t(n) \space y^2 = n $.

Consider the diophantine $$ x^2 + t(n) \space y^2 = n $$ Where $x,y> 0$, $ n>1$ and $t(n) > -1$. For a given $n$ we want to find the smallest possible integer $t(n)$. Clearly when $n$ ...
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341 views

$x^2-(p^4-16)y^2=p^2 \land x \mod (p^2-4) =2 \land x >0 \implies p | x,y$?

I wish to have a proof of $$x^2-(p^4-16)y^2=p^2 \land x \mod (p^2-4) = 2 \land x>0 \implies p | x,y$$ for primes $p$. It is easily proved that for a counter-example, $p$ must be $\equiv 1 \mod 4$:...
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Integer solutions of $3^n-1=2m^2$

I want to find solutions of $3^n-1=2m^2$ other than $(n, m)=(0, 0)$, $(n, m)=(1, \pm1)$, $(n, m)=(2, \pm2)$ and $(n, m)=(5, \pm11)$. There are no other "small" solutions ($n<1000$). For even $n$, ...
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1answer
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Non solubility of a Pell equation

I have this Pell equation $$x^2-223y^2=-3$$ I know there aren't solutions over the integers. How can i prove it without using things like continuous fractions? I tried reducing modulo some prime but I ...
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77 views

Solve $x^2+(8y)^2=p^2(4p^2y^2+1)$

I am trying to find solutions for $x^2+(8y)^2=p^2(4p^2y^2+1)$ for integer $x,y$ where $p$ is a prime $\equiv 1 \mod 4$ that does not divide $x,y$. I think there are no solutions but I could not prove ...
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Generalized Pell's equation

We know that Let $d$ be a positive square free integer and $r$ an integer satify $r^2+|r|\le d$. Suppose $x$ and $y$ are positive integers that satify $x^2-dy^2=r$. Then $\frac xy$ is a convergent to ...
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Can $n+1$ , $2n+1$ , $3n+1$ all be perfect squares , if $n$ is a positive integer?

Can the numbers $n+1$ , $2n+1$ and $3n+1$ be simultaneously perfect squares for any positive integer $n$ ? I tried to find that out and arrived at the equation system $$c^2-3a^2=-2$$ $$b^2-2a^2=-1$$ ...
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How can I find the complete set of fundamental solutions of a pell-like equation?

Let $D>1$ be a non-square-integer and assume that $$x^2-Dy^2=k$$ with integer $k$ has a solution with integers $x,y$. How can I find the complete set of fundamental solutions , if I know one ...
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If a solution exists, how can I construct another solution with coprime integers?

Suppose , $n$ is an odd positive integer and positive integers $a,b$ not being coprime satisfy the equation $$a^2-2b^2=n$$ Can I always find a solution of $$c^2-2d^2=n$$ with coprime integers $c,d$ ...
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$x^2-Dy^2=-1$ when D is square-free and has only prime divisors $\equiv 1 \mod 4$

I wish to know whether the Pell equation $x^2-Dy^2=-1$ can have any solutions when $D>0$ is a square-free product of primes $\equiv 1 \mod 4$ only. If there are any possible D for which this is the ...
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A System of Simultaneous Pell Equations

Are $0$ and $\pm1$ the only integer solutions for which both $\sqrt{24n^2+1}$ and $\sqrt{48n^2+1}$ are simultaneously integers ? Whilst pondering upon the Biblical concept of the Jubilee year, I ...
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Are the sets $\mathbb X=\{0,1,4,15,56,…,x_h,…\} $ and $\mathbb Y=\{0,2,12,70,408,…,y_i,…\}$ (excepting the elements $x_0=y_0=0$) disjunct?

In trying to answer another question I came to the problem as written in the title: Q1: Are the sets $\mathbb X=\{0,1,4,15,56,...,x_h,...\} $ and $\mathbb Y=\{0,2,12,70,408,...,y_i,...\}$ (...
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166 views

Fundamental Solution of Pell's Equation

To find the solutions of Pell's equation $x^2 - dy^2 = 1$, one can look at the convergents of the continued fraction expansion of $\sqrt d$: If $(x, y)$ is a non-trivial solution, then $x \over y$ is ...
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81 views

algebra direct connect pell eqn soln $(p_{nk},q_{nk})$ with $(p_n + q_n\sqrt{D})^k$

Apologies if this question has been asked before - there are many "pell" entries in this forum. Given: (a) $\;D \in \mathbb{Z^+}\;$ is not a perfect square. (b) The continued fraction expansion of $...
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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 ...
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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 ...
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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 ...
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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 $\mathbb{Z}$? ...
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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 ...
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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 ...
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1answer
434 views

Solutions for $65x^2-57y^2=8 \cdot 61$

I am looking for solutions of $65x^2-57y^2=8 \cdot 61$. I think there are none but I am unable to prove this. Rewriting the equation to $$57(x+y)(x-y)=8(61-x^2)$$ implies (among other things) that $...
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1answer
76 views

how to geometrically explain why pell numbers close to sqrt 2

How to graphically explain that the limit of yn/xn = $\sqrt 2$, as n approaches to infinity? Like, I know how to prove it algebraically.
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2answers
109 views

Finding integer solutions to $n^2 = 2d^2 - 2d + 1$

I'm trying to find the smallest integer solution to $$n^2 = 2d^2 - 2d + 1$$ Additional constraints: $$d > 10^{12}, n > 0$$ I wrote a computer program to bruteforce it, but that is too slow. ...
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2answers
51 views

Derive Closed Solution From Two Linear Recurrences

Let $x_0 = y_0 = 1$ and for $n \ge 1$ $x_n = 3x_{n - 1} + 4y_{n - 1}, \\ y_n = 2x_{n - 1} + 3y_{n - 1}.$ Goal. Determine an explicit formula for $x_n$ or $y_n$ just depending on $n$. Background. ...
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2answers
106 views

Generalizing the Pell equation $x^2-9\times89y^2 = 1$?

I. The fundamental solution to the Pell equation, $$x^2-61y^2=1$$ will stand out being "largish" as it is the $\color{blue}{6\text{th}}$ power of a fundamental unit $U_d$, $$(U_{61})^6 = \big(\...