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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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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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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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 (ax)^2-aby^2 = ...
1 vote
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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 ...
1 vote
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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 ...
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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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