Questions tagged [pell-type-equations]
Questions on finding integer solutions to bivariate equations of the form $x^2-Dy^2=a$.
357
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To get all the (b,a,a+1) Pythagorean triplets [closed]
In $(b,a,a+1)$
I found that $a+a+1=b^2$, in which $a>b$
You guys probably knew this but I randomly thought this thing in $5,12,13$ and then found it in all with $b,a,a+1$
So anyways
That means $2a+...
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Solvability of this General Pell's Equation
If $p\equiv -1 \pmod{12}$ is a prime, does this general Pell's equation always have integer solutions for $x$ and $y$?
$$x^2-3y^2=-p$$
I have tested that this is true for all values of $p$ up to $8\...
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3
answers
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How do I find another generalized Pell equation solution after finding the fundamental solution?
If I have a solution $(x_0, y_0)$ to $x^2 - Dy^2 = k$ where D > 1 is not a square, how do I find another solution $(x_1,y_1)$ to $x^2 - Dy^2 = k$ without knowing the solution $(u,v)$ to $u^2 - Dv^...
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System of Pell's equations
Let $a$, $b$ and $k$ be positive integers.
I want to prove that the only solution of the system
$$\left\{\begin{array}{rclr}
2k^2+1&=&a^2&(1)\\
6k^2+1&=&b^2&(2)
\end{array}
\...
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votes
1
answer
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Missing pattern in solvable negative Pell equation
Considering the negative Pell equation $ x^2 - Dy^2 = -1 $, I know that a necessary condition for solvability is that
$ D = a^2 + b^2$, with $ a,b $ positive integers.
If I fix $ b = 1 $, NPE is ...
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Analytic Number Theory - distribution of $x^2$ vs. the distribution of $x^2 - 2y^2$
My question originates from Rational Points on Elliptic Curves, (Silverman & Tate), though has little to do with elliptic curves.
In chapter $V$: Integer Points on Cubic Curves, section $3$ it ...
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Command for negative pell equations in PARI-GP or SAGE [closed]
Is there any common command for testing negative Pell equations in PARI-GP or Sage?
I searched for this question online and realized that
Testing negative Pell equations online is checking solubility, ...
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1
answer
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Is it possible to further generalise Brahmagupta's identity?
Is it possible to generalise Brahmagupta's identity
further, by which I mean, for different n, for example take the equations:
$$1357 = 37^2 - 3\times2^2$$
$$1357 = 38^2 - 87\times1^2$$
$$1357^2 = ...
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Solve Diophantine equation $a^2+5ab+3b^2-c^2=0$
Solve Diophantine equation $a^2+5ab+3b^2-c^2=0$
My thoughts are to express it as $(pa+qb)^2 = c^2 $and then solve it as Pell's equation.
One solution is $(1,9,17)$.I don't know whether it is a ...
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How to find the general term for Pell's sequence with Euler's method
Hello i try to find the general term for the Pell's sequence with this method called Euler's method .
$$
a_{n+1}=2a_{n}+a_{n-1}
$$
What I tried here is :
$$G(x)=\sum_{n=0}^{\infty }a_{n}x^{n}$$
And i ...
1
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1
answer
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Integral solutions to $A^2=(u^2+uv+2)/2$, $B^2=(v^2+uv+2)/2$, $C^2=1+uv$
The system of equations $A^2=(u^2+uv+2)/2$, $B^2=(v^2+uv+2)/2$, $C^2=1+uv$ has $A,B,C,u,v$ all integral.
Numerical evidence found by brute force computation implies that $uv(u^2-14uv+v^2-16)=0$.
...
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Given $d$, how many values of $n$ should I test to get a square of form ${2n^2+d}$
Given $d$, how many values of $n$ should I test to get a square of the form ${2n^2+d}$
Both $d$ and $n$ are a positive integers.
There must also be some periodicity in $n$ to jump from the first ...
user1057393
2
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1
answer
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For solving a Pell's equation How many iterations should I run before confirming no solution exists.
I am running this piece of code from rosettacode
...
user1092017
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0
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Solvability of $ax^2 +b = y^2$ under integer, and has infinite solutions.
For example,
$3x^2 + 4 = y^2$ has infinite integer solutions while $5x^2 + 12 = y^2$ doesn't.
At first I thought $b$ has to be the perfect square but $3x^2 + 6 = y^2$ has infinite solutions.
So the ...
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0
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Finding a representation $N = x^2 + ny^2$ efficiently
I want to find a representation of an integer $N$ of the form $N = x^2 + ny^2$. Any single non-trivial representation (i.e. $x \gt 0, y \gt 0, n \ge 1$) suffices.
A naive algorithm that I have ...
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System of two Pell's equations that might have infinitely many solutions
Does there exist infinitely many integers $(a,b,c)$ verifying simultaneously the two following equations:
$$a^2-2b^2=-1$$ and
$$a^2-24c^2=1$$
Well I tried this problem with Pell's equation approach $a=...
user860474
4
votes
2
answers
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Finding a solution to the generalised Pell's equation $x^2-31y^2=5$
I'm trying to find a solution to the generalised Pell's equation $x^2-31y^2=5$. So far I have found the fundamental solution $x=1520,y=273$ to the equation $x^2-31y^2=1$, obtained from calculating the ...
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2
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Integer Solutions to the equation $x^6+5y^2=z^2$
This problem is on Page 194 of “Number Theory and Its History”, by Oystein Ore. I can find infinitely many solutions by letting $x=1$ so that $z^2-5y^2=1$ which has infinitely many solutions. So my ...
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1
answer
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What's the inverse element in the group of solutions to Pell's equation?
I'm working on a problem from a past exam paper,
Define the group operation on the set of solutions of Pell's equation, $$G_d=\{(x,y)\in\mathbb{Z}^2:x^2-dy^2=1\}$$ and show that the group axioms are ...
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Is there a short proof that the set of solutions to Pell's equation form an abelian group?
I'm working on a question from a past exam paper,
Show that the set of solutions $G_{d,p}$ to Pell's equation $x^2-dy^2=1$ modulo $p$ is a finite abelian group, and compute the order of this group ...
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Positive integer solutions of $ab+1=x^2, ac+1=y^2, bc+1=z^2, x+z=2y$
A question essentially the same a this one was asked in
MSE 4479792
without any background details and was deleted by the post author
after I posted a minimal one sentence answer mentioning five OEIS
...
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fundamental solution of Pell’s equation under special conditions
I make the following conjecture from an answer.
conjecture
Suppose $N$ could be expressed as
$$
N= p^2\pm q,\ \ \ \ (p,q \in \mathbb{N},\ 1 \lt q \lt p,\ q\mid 2p)
$$
In this case, the fundamental ...
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Solution to Pell's equation $a^2 - Nb^2 = 1$ when $a$ is an even integer
Given an integer $a$, show that there is a square-free integer $ N > 1$ and an integer $b$ such that $a^2 - Nb^2 = 1$.
I'm not too sure how to go about proving this. I believe I'm supposed to use ...
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How is the Pell-like equation $ax^2-by^2 = c$ solved?
Suppose $a, b$ and $ab$ are non-square and solution exist for the above equation. One way we can solve it is by multiplying $a$ to the equation to get the Pell equation
\begin{equation}
(ax)^2-aby^2 = ...
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0
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Pell's equation fundamental solution vs. an earlier convergent of $\sqrt{d}$
Let $(p,q)$ be the fundamental solution to Pell's equation $x^2-dy^2=1$, which makes $\frac{p}{q}$ a convergent of $\sqrt{d}$, per the theorem underlying the continued fraction algorithm for solving ...
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Is there a solution (x,y) to a Pell equation such that $x\equiv \pm 1 \; \mathrm{mod} \; y$?
I have the following question.
Let $x^2-Ny^2=1$ be a Pell equation (where $N$ is not a square).
Is it possible to find a solution $(x,y)$ such that $x\equiv \pm 1 \; \mathrm{mod} \; y$?
If there ...
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2
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Perfect numbers and Pell's equation
(Disclaimer: The following is a naive attempt to apply the theory of Pell's equations to perfect numbers. Please bear in mind that this is my first time to try solving such an equation.)
Let $p^k$ be ...
1
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2
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Generalized Pell's equation where $N$ is perfect square
Are all solutions of the equation $x^2-4My^2=K^2$, multiples of $K$? I am considering $M$ not perfect square. Any tests in Python show be true, but...
My code:
...
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Binary quadratic function, when it is equal to a square number?
$x^2+5xy+3y^2=T$ and $\gcd(x,y)=1$
T is a square number. I need to find which x and y values make the function a square number. Gcd is a greatest common divisor function.
I converted the function Into ...
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Notice that $\sqrt{51}\approx 7+\frac{\sqrt{2}}{10}$
I was doing some other stuff and noticed that:
$$\sqrt{51} - \left(7 + \dfrac{\sqrt{2}}{10}\right) = 0.0000070723\,\, (*)$$
and this immediately made me think of their respective continued fractions, ...
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Problem (perhaps trivial) with the continued-fraction approach to (generalized) Pell-solutions
I've seen some Q&A's here which put light on the continued-fractions approach to finding solutions for the Pell-equation. I'm trying to get familiar with this method to apply it to my style of ...
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Solutions to the Pell equation $(2x+y)^2-5y^2=4$
The Pell equation $(2x+y)^2-5y^2=4$ has fundamental solution $(1,1)$. I need to find three more solutions, I know tha they are the Fibonacci numbers in ordered pairs $(2,3), (5,8),...$, but I don't ...
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Do I catch all solutions for the generalized Pell-equation $a^2+b^2 = 2 c^2$ by this matrix-method?
As one question in a somewhat bigger analysis I want to characterize the set of solutions of the generalized Pell-equation in the title:
$$a^2+b^2 = 2 c^2 \tag 1$$
I'm not much fluent with the Pell-...
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Fourth Degree Pell Equation
I got stuck at the following of my research problem:
Prove that only solution to equation $4b^2-3a^4=1$ for odd positive integers $a$, $b$ is $(1,1)$.
I made factorization -->$3a^4 = (2b-1)(2b+1)$...
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Proving $\frac{L_n(x)^2}{4}-\left(\frac{x^2}{4}+1\right) \cdot F_n(x)^2=(-1)^n$, where $L_n(x)$ and $F_n(x)$ are the Lucas and Fibonacci polynomials
Recently, I found the following identity:
$$\frac{L_n(x)^2}{4}-\left(\frac{x^2}{4}+1\right) \cdot F_n(x)^2=(-1)^n $$
where $L_n(x)$ denotes the Lucas polynomials and $F_n(x)$ denotes the Fibonacci ...
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A general formula for the solutions of the negative Pell's equation
I want to find an explicit formula for the solutions of the negative Pell's equation
$x^2 - Dy^2 = -1$ in terms of {a, b} with $D = a^2 + b^2$
I already know that the nth solution is given by the ...
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When do there exist rational solutions to the generalized Pell equation?
There are many questions on this site asking about integer solutions to the generalized Pell equation $x^2 - dy^2 = n$ for $d$ and $n$ integers and $d$ squarefree. What is known about the existence of ...
user323867
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$\lfloor \sqrt{D}\rfloor>q_{l-1}\implies$no odd solution to $x^2-Dy^2=\pm4$?
I try to show that $a_0=\lfloor \sqrt{D}\rfloor>q_{l-1}$ (or $\frac{q_{l-1}}{2}$ if $q_{l-1}$ is even) implies that there is no odd solution to the Pell equation $$x^2-Dy^2=\pm4$$
$D$ is a positive ...
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Solve in $\mathbf{N}$ the equation $9x^2+p=y^2$
In these days, I have been trying to solve this problem:
Let $p \in \mathbf{N}$ a positive large integer ($> 10^9$). Find all $x, y \in\mathbf{N}$ such that:
$$9x^2+p=y^2$$
The first approach ...
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$p$-adic structure of Pell-type equations
As an example, consider an integer solution of $ x ^ 2-3y ^ 2 = 13 $.
$ y $ that satisfies this equation is
$y_k = \frac {(4+ \sqrt {3}) (2+ \sqrt {3}) ^ k-(4- \sqrt {3}) (2- \sqrt {3}) ^ k} { 2 \sqrt ...
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Seeking examples of proofs [by contradiction?] using Pell equation
I’m looking for proofs which use Pell equations as a critical part of the proof. One example is this paper by Robert Phillips: on page 4, he says
If $(p, q)$ is one of the infinite number of ...
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Solving $X^2-6Y^2=Z^3$ in positive integers
I’m trying to solve the Diophantine equation
$$X^2-6Y^2=Z^3 \tag{$\star$}$$
in positive integers $x,y,z$.
Brute force calculations confirm the naïve intuition that there are many [read: surely ...
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votes
1
answer
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When does pell equation not have infinite solution?
I am working on a problem and while trying to make some progress, I stumbled across a sub-question.
When does $x^2-2y^2 = n$ NOT have infinite solutions for some positive integer $n$.
Now I tried to ...
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Given $k$, can we find $n$ such that $d$ is a perfect square
Given an odd positive integer $k$. Define $$d=16(k+1)^3(k+2)(n+1)^2+1$$ where $n$ is alos a positive integer.
My question is: Given $k$, can we find $n$ such that $d$ is a perfect square.
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Find all $q \in \mathbb{N}$ such that $\frac{q(q+1)}{12}$ is a Perfect square.
Find all $q \in \mathbb{N}$ such that $\frac{q(q+1)}{12}$ is a Perfect square.
Trivially we see that $q=3$ is the first candidate.
Now let $$\frac{q(q+1)}{12}=r^2$$
$\implies$
$$q^2+q-12r^2=0$$
By ...
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Looking for a book/proof for Pell's equation $x^2-2y^2=-1\ \Leftrightarrow\ x+y\sqrt{2}=(1+\sqrt{2})^k$ with odd k
I got this Lemma to Pell's Equation
$$
x^2-2y^2=-1\ \Leftrightarrow\ x+y\sqrt{2}=(1+\sqrt{2})^k\ \text{with k odd}\\
x^2-2y^2=+1\ \Leftrightarrow\ x+y\sqrt{2}=(1+\sqrt{2})^k\ \text{with k even}
$$
...
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0
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Patterns in convergents of continued fraction of $\sqrt{D}$?
First, to give some background: If $D$ is an integer, then the continued fraction of $\sqrt{D}$ is always periodic. For example, the continued fraction of $\sqrt{7}$ is $[2; \overline{1,1,1,4}]$. Also,...
1
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1
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Find the $d$ such $(d^2+d)x^2-y^2=d^2-1$ has postive integer $(x,y)$
Find all the positive integers $d$ such that
$$(d^2+d)x^2-y^2=d^2-1$$ has a positive integer solution $(x,y)$.
maybe use Pell equation some result
$$x^2-Dy^2=C$$ to solve it?
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Nice upper bound on $max{|x|, |y|}$ in terms of $d$ in Pell's Equation
Given Pell's equation $x^2 - dy^2 = 1$, can we come up with a nice upper bound on $max(|x|,|y|)$ in terms of $d$? Also, if it's somehow more tractable, are we able to find an upper bound on $|x+y|$? ...
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4
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General term for series $0,2,12,70, 408...$
Given the curves $y=\sqrt{2x^2+1}$ and $x=\sqrt{2y^2+1}$, the first line (line with smallest y coordinate) which intersects the curves with integer coordinates is $x+y=1.$ This line intersects the ...
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