# 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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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 $u^2 - Dv^... 6 votes 1 answer 208 views ### System of Pell's equations Let$a$,$b$and$k$be positive integers. I want to prove that the only solution of the system $$\left\{\begin{array}{rclr} 2k^2+1&=&a^2&(1)\\ 6k^2+1&=&b^2&(2) \end{array} \... 2 votes 1 answer 144 views ### Missing pattern in solvable negative Pell equation Considering the negative Pell equation x^2 - Dy^2 = -1 , I know that a necessary condition for solvability is that D = a^2 + b^2, with a,b positive integers. If I fix b = 1 , NPE is ... 0 votes 0 answers 103 views ### Analytic Number Theory - distribution of x^2 vs. the distribution of x^2 - 2y^2 My question originates from Rational Points on Elliptic Curves, (Silverman & Tate), though has little to do with elliptic curves. In chapter V: Integer Points on Cubic Curves, section 3 it ... 1 vote 2 answers 91 views ### Command for negative pell equations in PARI-GP or SAGE [closed] Is there any common command for testing negative Pell equations in PARI-GP or Sage? I searched for this question online and realized that Testing negative Pell equations online is checking solubility, ... 2 votes 1 answer 110 views ### Is it possible to further generalise Brahmagupta's identity? Is it possible to generalise Brahmagupta's identity further, by which I mean, for different n, for example take the equations:$$1357 = 37^2 - 3\times2^21357 = 38^2 - 87\times1^21357^2 = ... -1 votes 2 answers 154 views ### 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 ... 0 votes 1 answer 43 views ### 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 vote 1 answer 95 views ### 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$uv(u^2-14uv+v^2-16)=0$. ... 4 votes 2 answers 123 views ### 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 ... 2 votes 1 answer 137 views ### 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 ... 0 votes 0 answers 55 views ### 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 ... 1 vote 0 answers 46 views ### 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 ... 1 vote 0 answers 89 views ### 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=... 124 views

### 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 ...
205 views

### 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 ...
255 views

### 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 ...
1 vote
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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 ...
310 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 = ...
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 ...
1 vote
241 views

### 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
141 views

### 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: ...
71 views

### 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, ...
1 vote
108 views

### 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 ...
1 vote
134 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 ...
302 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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### 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 ...
1 vote
283 views

### 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 ...
136 views

### 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 ... 166 views

### $\lfloor \sqrt{D}\rfloor>q_{l-1}\implies$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 ...
### 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 ...