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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Pell-Type Equation: show that the solutions $x,y$ are such that $\frac{x}{y}$ is a convergent

Prove that for any solution of the equation $x^2 − 14y^2 = 2$, in positive integers $x, y$ the value $\frac{x}{y}$ is a convergent of $\sqrt{14}$ I am not too sure where to start, I have the continued ...
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Binary quadratic forms over Z of discriminant D [closed]

For each of the following 3 discriminant D = 4d, how can I classify all binary quadratic forms over Z of discriminant D. Is there a way to describe the river corresponding to the canonical form Q(x, y)...
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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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Chebyshev polynomial and some perfect squares

Let $U_{n}(x)$ be the Chebyshev polynomial of the second kind (https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html). I am asking if the following quantity is still a perfect square: $...
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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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-1 votes
1 answer
69 views

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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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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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 ...
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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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4 votes
2 answers
160 views

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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1 answer
106 views

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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185 views

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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2 votes
2 answers
147 views

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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1 answer
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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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0 answers
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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 ...
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Estimating the number of solutions for a quadratic inequality.

It is well known in the literature that if $0< A, E < N^{O(1)}$ ($A, E$ integers) and $A$ is not a perfect square, then the inequality $$ \left|x^2 + A y^2 - E\right| \leq \frac{1}{2} \qquad (\...
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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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5 votes
5 answers
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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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6 votes
2 answers
174 views

$p$-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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4 votes
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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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12 votes
8 answers
486 views

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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2 votes
1 answer
110 views

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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-3 votes
1 answer
120 views

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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3 votes
2 answers
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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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2 votes
1 answer
64 views

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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4 votes
0 answers
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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,...
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2 votes
1 answer
311 views

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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1 vote
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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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2 votes
4 answers
120 views

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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2 votes
2 answers
155 views

Showing that Pell's equation has at least one integer solution

I'm studying Pell's equations of the form $x^2-dy^2=1$ for $d$ a square-free natural number and $x,y$ integers. In particular, I want to show that such an equation always has a solution. I see that ...
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5 votes
4 answers
235 views

A caboodle of Pell's equation in one? $x^2+y^2-5xy+5=0$

I saw this twitter post that reads: Find all the pairs of positive integers $(x,y)$ satisfying $$ x^2 + y^2 - 5xy + 5 = 0 . $$ I don't know how to tackle this and I ended up summoning WolframAlpha ...
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0 votes
2 answers
222 views

Find all the integer solutions of $x^2-4y^2=1$

Find all the integer solutions of the equation: $x^2-4y^2=1$. I know I can't solve it like a PELL equation because d is a square in this case. Would appreciate your help:)
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18 votes
2 answers
668 views

Primes in solutions to Pell-type equations

What is known about primes in solutions to Pell-type equations? In particular, consider the negative Pell equation $x^2 - 5 y^2 = -1$. As far as I've been able to check (in the first $4000$ solutions) ...
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2 votes
2 answers
348 views

An infinite family of generalized Pell equations

Let $m,n,x,y,z$ be positive integers such that $y=m(z-1)+1$ and $yz-1=n(y^2-x^2)$. If you fix $m$ and $n$ then this is a generalized Pell equation in $x$ and $z$ (i.e., a quadratic Diophantine ...
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5 votes
1 answer
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Difficult implications of a detail in an(other) approach to the simultaneous Pell equations $24a^2+1=t^2$ and $48a^2+1=u^2$

In reviewing & improving my attempt to solve the question on simultaneous Pell-equations (see here in MSE) I came across a detail which has surely wider implications and which I cannot encompass (...
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3 votes
2 answers
130 views

Solving $\tau^4 +1=u^2 +v^2$ over the integers

I'm trying to show, that the solutions $t_i$ from Pell's equation $t^2-2s^2=\pm 1$ are not squares (only for the trivial cases). For the even solutions $t_{2i}$ (solutions for the + sign) this is easy,...
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0 votes
1 answer
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How to generate infinitely many solutions to $x^2-dy^2 = 4$?

I was looking at how to generate infinitely many solutions to the Pell's equation $x^2-dy^2 = 1$, where $d$ is a square-free positive integer. On https://en.wikipedia.org/wiki/Pell%27s_equation#...
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3 votes
4 answers
326 views

Are there Pell Equations $x^2 - dy^2 = n$ that are easy to solve?

Consider the Pell equation $x^2 - dy^2 = n$ where $d$ is a positive non-square integer. Are there examples of special $d$ that makes it easy to solve (obtain non-trivial solutions) the equation for ...
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1 vote
1 answer
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Is every integer $z$ representable in Pell form as $x^2 \pm dy^2 =z$?

We know that there are integers that cannot be represented as the sum of two squares (Fermat's Four Square Theorem). We also know that every natural number can be represented as the sum of four ...
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0 votes
1 answer
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Method for solving Diophantine equation $ax^2 + bx + c = y^2$

How do I solve the Diophantine equation $ax^2 + bx + c = y^2$? The approach I have so far is to use the transformation $X = 2ax + b$ and $Y = 2y$. Applying this, we get, $X^2 - dY^2 = n$, where $n = b^...
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0 votes
4 answers
138 views

Square Triangle numbers and Pell's equation

So i have just started to study number theory and i was asked this question. Now i tried to search online and i found out pell equation can used to solve this question. Now in an online video i saw ...
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2 votes
1 answer
123 views

Infinitely many solutions of the equation $\frac{x+1}{y}+\frac{y+1}{x} = 4$ [closed]

Prove that there exists infinitely many positive integer solutions in $(x,y)$ to the equation : $$\frac{x+1}{y} + \frac{y+1}{x} = 4$$
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2 answers
133 views

Recurrence relation solution to negative Pell's equation

I'm trying to determine how to get the sequence of possible solutions for a negative Pell's equation: $$ x^2 - 2y^2=-1 $$ I know that the fundamental solution is $x_1=1$ and $y_1=1$, but I don't know ...
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1 vote
1 answer
42 views

The Integral Values For Which The Expression Is A Perfect Square

Given that x is an integer, when does the expression $4x^2 + 80$ form a perfect square? I tried putting $x=4$, got a perfect square but i am not able to calculate how many such cases can be there? ...
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1 vote
2 answers
70 views

Question about Pell's equation.

Question. Suppose that $x,y,x',y'$ are positive integers satisfying $x^2-dy^2=\pm 1$ and $(x')^2-d(y')^2=\pm 1$ respectively. Assuming $x<x'$, prove that $y<y'$. Not too sure where to begin. ...
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1 vote
1 answer
129 views

A problem involving the largest prime divisor of polynomials of the form $a^2+1$

Question: Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that there exist infinitely many positive integer tuples $a, b, c$ ( all three distinct) such that $P(a)=P(b)=P(c)$. I ...
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