Linked Questions

15
votes
6answers
2k views

How to find solutions of $x^2-3y^2=-2$?

According to MathWorld, Pentagonal Triangular Number: A number which is simultaneously a pentagonal number $P_n$ and triangular number $T_m$. Such numbers exist when $$\frac{1}{2}n(3n-1)=\frac{1}{...
13
votes
5answers
1k views

Another quadratic Diophantine equation: How do I proceed?

How would I find all the fundamental solutions of the Pell-like equation $x^2-10y^2=9$ I've swapped out the original problem from this question for a couple reasons. I already know the solution to ...
20
votes
2answers
2k views

Find all integer solutions for the equation $|5x^2 - y^2| = 4$

In a paper that I wrote as an undergraduate student, I conjectured that the only integer solutions to the equation $$|5x^2 - y^2| = 4$$ occur when $x$ is a Fibonacci number and $y$ is a Lucas number. ...
6
votes
6answers
2k views

Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares

How many positive integer $n$ are there such that $2n+1$ , $3n+1$ are both perfect squares ? $n=40$ is a solution . Is this the only solution ? Is it possible to tell whether finitely many or ...
8
votes
5answers
371 views

If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.

I came across this question on another forum. The question is: $$ \text{If $m,n\in \mathbb{Z}_+$ such that $3m^2+m=4n^2+n$, then $(m-n)$ is a perfect square.}$$ I have managed to partially prove ...
8
votes
4answers
1k views

Does the Pell-like equation $X^2-dY^2=k$ have a simple recursion like $X^2-dY^2=1$?

If $d \ne 0$ is a non-square integer, and $(u,v)$ is an integer solution to the Pell equation $$ X^2 - dY^2 = 1, \tag{$\star$} $$ then each solution $(x_i,y_i)$ can be recursively calculated using ...
16
votes
3answers
243 views

Is there a simple proof that if $(b-a)(b+a) = ab - 1$, then $a, b$ must be Fibonacci numbers? [duplicate]

Consider the identity $(b-a)(b+a) = ab - 1$, where $a, b$ are nonnegative integers. We can also express this identity as $a^2 + ab - b^2 = 1$. This identity is clearly true when $a = F_{2i-1}$ and $...
1
vote
5answers
880 views

Generate solutions of Quadratic Diophantine Equation

Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: The quadratic diophantine $ k^2 - 1 = 5(m^2 - 1)$ Here's the answer: ...
5
votes
2answers
1k views

Finding all solutions of the Pell-type equation $x^2-5y^2 = -4$

I wanted to solve the equation $x^2-5y^2 = -4$ with $x$ and $y$ integers. Let $\omega=\frac{1+\sqrt5}{2}$ and $A = \mathbb{Z}[\omega]$. One can reduce the Pell equation to finding the elements of $A$ ...
2
votes
2answers
597 views

Solving the equation $ x^2-7y^2=-3 $ over integers

I'd like to solve the following Pell equation: $$ x^2-7y^2=-3 $$ Where $x$ and $y$ are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ...
5
votes
2answers
271 views

Maps of primitive vectors and Conway's river, has anyone built this in SAGE?

I am attempting to teach number theory from John Stillwell's Elements of Number Theory in the upcoming semester. There are two sections (5.7 and 5.8) which describe the diagrammatic method for the ...
3
votes
1answer
716 views

how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$

solve $ 3x^2+3xy-5y^2=55$ using number theory tools ,i have found the following $\Delta=3^2+4(5)(3)=9+60=69$ $d=69,u=1$ $w_{69}=\frac{1+\sqrt{69}}{2}$ $O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}...
5
votes
1answer
306 views

Why can't the Alpertron solve this Pell-like equation?

Dario Alpern's Alpertron is convenient for solving Pell and Pell-like equations. It can even solve the one at the heart of Archimedes' cattle problem, $$p^2-(4)(609)(7766)(4657^2)q^2=1$$ and give ...
6
votes
2answers
121 views

Solve the following equation for x and y:

$x^2 = y^2 + xy + 5$, where $x$ and $y$ are natural numbers. Here is what I have so far: $x \neq y$ (from the equation). $x$ is always odd (using the equation and assuming $2$ cases - $y$ is odd or $...
1
vote
1answer
134 views

To find all integral solutions of $3x^2 - 4y^2 = 11$

I have to find all integral solutions of $3x^2 - 4y^2 = 11$. I looked at the equation $\text{mod}\ 3$, and $\text{mod}\ 4$ to see that $x^2 \equiv 1\ \text{mod}\ 4$ and $y^2 \equiv 1\ \text{mod}\ 3$ ...

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