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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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 ...
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1answer
37 views
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 ...
0
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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
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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 ...
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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 ...
5
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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 ...
2
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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
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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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0answers
32 views
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
$$\mathcal{P}_d(\mathbb{Z})^+=\lbrace(x_0,y_0)\in\mathbb{Z}^2_{\geq1}:x_0^2-dy_0^2=1\rbrace\neq\...
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1answer
25 views
pell's equation converge
Explain Why pell's equation $x_n+ny_n=1$, $(x_n/y_n)^2$ is coverge to n as n increase
For example, n=11 the answer $(x_n/y_n)^2$ is very close to 11 when n increase.
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0answers
50 views
Conditions on those Pell-type equations to admit solutions
I am facing those generalised Pell equations,
$a^2-Db^2=-8$
and
$x^2-Dy^2=8$,
where $D=t^2+8t$ for an odd $t$, $t>2$. In particular, I would like to find the cases where both equations admit (...
6
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2answers
139 views
Evaluating $\liminf_{n\to\infty}n\{n\sqrt2\}$
How can we evaluate $$\liminf_{n\to\infty}n\{n\sqrt2\},$$where $\{\cdot\}$ denotes the fractional part of $\cdot$?
The first thing came to my mind is Pell's equation $x^2-2y^2=1$.
Knowing that $\...
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5answers
114 views
If $2n+1$ and $3n+1$ are perfect squares, then prove that $8|n$.
If for some number $n\in \mathbb N$, the numbers $2n+1$ and $3n+1$ are perfect squares of integers, then prove that $8|n$.
if $2n+1=m^2$ and $3n+1=k^2$ then $k^2-m^2=3n-2n+1-1=n$ now I need to show ...
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3answers
56 views
How do i compute $f_n = 3f_{n-1} + 2\sqrt{2f_{n-1}^2 - 2}$ for $n$ around $10^{18}$?
So I have the recurrence $$f_n = \begin{cases}
3f_{n-1} + 2\sqrt{2f_{n-1}^2 - 2}, &n > 1\\
3, &n = 1\\
\end{cases}$$
and I need to compute it in $O(\lg n)$, for $n$ as big as $10^{18}$.
I ...
5
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1answer
77 views
Has my conjecture using Pell's Equation been discovered before
We have $$\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)^...
1
vote
1answer
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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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 $\sqrt{d}.$
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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 ...
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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 $$\frac{(3+2\sqrt2)^n+(3-2\sqrt2)^n}{2}$$ ...
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2answers
259 views
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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5answers
117 views
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 ...
3
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1answer
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 ...
5
votes
1answer
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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0answers
30 views
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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0answers
73 views
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 $n$ ...
7
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2answers
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 $\equiv 1 \mod 4$:...
6
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2answers
130 views
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 $n$, ...
2
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1answer
37 views
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 ...
3
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1answer
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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1answer
125 views
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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1answer
224 views
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 $$c^2-3a^2=-2$$ $$b^2-2a^2=-1$$ ...
3
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2answers
191 views
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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vote
2answers
60 views
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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3answers
74 views
$x^2-Dy^2=-1$ 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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5answers
631 views
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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1answer
119 views
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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1answer
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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1answer
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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2answers
64 views
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 ...
2
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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 ...
5
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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 ...
2
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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 $\mathbb{Z}$?
...
0
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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 ...
1
vote
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 ...
2
votes
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 $...
0
votes
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.
2
votes
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.
...
0
votes
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. ...
2
votes
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(\...
