# 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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### 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 $|x+y|$? ...
114 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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### 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 ...
149 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 ...
137 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:)
565 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) ...
327 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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### 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 (...
119 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,...
64 views

### 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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### 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 ...
70 views

### 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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### $x^n-dy^n$ irreducible?

I am dealing right now with a generalized form of the Pell Equation. In order to use Thue's Theorem I need to know that $x^n-dy^n$ ist irreducible over $\mathbb{Q}$. Somehow I don't get why. I tried ...
94 views

### Can anyone solve this Pell equation?

I have solved the Pell equation $p^2 - 95 q^2 =1$ . By looking at the convergents corresponding to the simple continued fraction of $\sqrt{95}$ I was able to find the fundamental solution $p=39$ ...
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### How to describe the solutions of a Pell equation which contains a rational number

Let $N(x,y) = x^{2}-dy^{2}$ with d a strictly positive integer and not a square and m an non-zero integer. $$N(x,y) = m$$ is the general form of Pell's equations which is mostly studied in the ...
62 views

### How many integers solutions does $x^2 + xy - y^2 = 1$ have?

Also, if we know if a binary form represents an integer $n$, is there an algorithm to find all the solutions?
90 views

### Integer solutions for $\frac{n(n-1)}2=m^2-1$

I want to find all integer solutions for $$\frac{n(n-1)}2=m^2-1.$$ The only ones I found are $m=\{2,4,64\}$ and $n=\{3,6,91\}$ meaning $m^2-1=\{3,15,4095\}$, but are they the only ones? If not, are ...
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### How to find every primitive solutions of a pell's equation ? And what if one solution is in a specific form?

For example, solve this : $x^2 +2x +3 = 10y^2+11y+12$ with x,y integers I can reduce this equation into this : $Y^2-40X^2=-279$ with $X=x+1$ and $Y=20y+11$ Now I need to find the primitive solutions ...
156 views

### IMO 1988 Q6 $a_n = …$

My question is in bold below. Here is a summary of my working on this problem which is question 6 from the IMO 1988 paper: For $\frac{a^2+b^2}{ab+1}=n$, where $a,b,n \in Z^+$ let $n=x^2$ then for ...
216 views

### Prove that the Pell's Equation $x^2 −Dy^2 = 1$ always has a solution where $y$ is a multiple of $41$

$D$ is a positive integer that is not a perfect square Recently I am taking a introductory number theory course and I met this question right after we learned Pell's equation and Diophantine ...
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### Integer solutions to the equation $7x^2 = y^2+y+1$

While investigating the related equation $7^n = m^2 + m+1,$ I managed to quite quickly handle the case that $n$ is even. If $n$ is odd, we may let $x = (n-1)/2.$ This now reduces to the question in ...
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### Pell's equation easy bound for the fundamental solution

Let $d>1$ be a non-square integer and consider $x^2 - dy^2 = 1$, with minimal solution $(x_0, y_0)$. Can anyone give an explicit bound for $x_0$ (or $y_0$ or $x_0 + y_0\sqrt{d})$ in terms of $d$, ...
181 views

### For $x^2-3y^2=1$ over integers more than $1$, can $\frac{y+1}2$ be square number?

For $x^2-3y^2=1$ over integers more than 1, can $\frac{y+1}2$ be square number? I know that $x^2-3y^2=1$ is one of pell's equation, so I know its general solution. But I know nothing about its ...
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### Complicated unusual Pell equation

I am trying to find the five smallest pairs of positive integers p,q that satisfy the unusual Pell equation $$p^2− \frac{251934169}{4} q^2 = 1.$$ One obvious trivial solution is p=1 and q=0 , but ...
I have been told that a simple linear transformation (or a change of variables) can transform the quadratic $$x^2+45xy-216y^2$$ into the Pell equation $$p^2−321q^2=1$$ However I have been unable to ...