Articles of pell type equations

How do you find the solutions for $(4n+3)^2-48y^2=1$?

I know that there are infinitly many because it is an Pell’s equations, but how do I find the first? I have seen that we have $(1,1)$. But by solving it with the formula for finding the solutions, I arrive at $n$ not whole… Could someone explain me how to arrive at $n$ whole without […]

How to solve a bivariate quadratic (not necessarily Pell-type) equation?

Simple Pell equations often have solutions that can be found with little work given certain conditions. These are of the form $x_{n}^{2} – A y_{n}^{2} = \pm 1$. There are harder equations that involve non-squared variable terms. In this view how is a solution to these equations found? As an example, what is the solution […]

For which primes $p$ does $px^2-2y^2=1$ have a solution?

Let $p$ be an odd prime. If $px^2-2y^2=1$ is solvable, we can get Jacobi symbol $(\frac{-2}{p})=1$, so $p=8k+1,8k+3$. But when $k=12$, $p=97$, the Pell equation $97x^2-2y^2=1$ is unsolvable. I think this diophantine equation is unsolvable for many integers. But it satisfies the Jacobi symbol, so my question is for what prime $p$, the diophantine equation […]

Solve $3x^2-y^2=2$ for Integers

If $x$ and $y$ are integers, then solve (using elementary methods) $$3x^2-y^2=2$$ I tried the following If $y$ is even, then $4|y^2$ and hence $2|y^2+2$ (and $4$ doesn’t divide it), but $3x^2=y^2+2$ and since for R.H.S. to be even, $2|x^2 \implies 4|x^2$, and we get a contradiction. So $x$ and $y$ both are odd. If […]

How to solve $(2x^2-1)^2=2y^2 – 1$ in positive integers?

I encountered this question (posed by Fermat) in a letter from Fermat to Carcavi and was wondering what would be the best elementary way to solve it. Solve in positive integers$$(2x^2-1)^2=2y^2 – 1$$ Any help will be appreciated. Thanks in advance.

When is $991n^2 +1$ a perfect square?

What should be the value of $n$ so that the number obtained after adding $1$ to $991$ times its square is itself a perfect square? Can you please give me a few hints on this topic with a few specific reasons?

Integer solutions to $ x^2-y^2=33$

I’m currently trying to solve a programming question that requires me to calculate all the integer solutions of the following equation: $x^2-y^2 = 33$ I’ve been looking for a solution on the internet already but I couldn’t find anything for this kind of equation. Is there any way to calculate and list the integer solutions […]

solve $x^2 – 25 xy + y^2 = 1$ does it have a solution?

Usually the Pell equation is written $x^2 – dy^2 = 1$ but here I am looking for solutions to an equation of the type: $$ x^2 – k xy + y^2 = 1 $$ and In particular, $k$ is a perfect square. So I am picking $k = 25$ an example. If we complete the […]

Rational solutions of Pell's equation

1) $D$ is a positive integer, find all rational solutions of Pell’s equation $$x^2-Dy^2=1$$ 2) What about $D\in\Bbb Q$ ?

Why does this test for Fibonacci work?

In order to test if a number $A$ is Fibonacci, all we need to do is compute $5A^2 + 4$ and $5A^2 -4$. If either of them is a perfect square, the number is Fibonacci, otherwise not. Why does this test work? Is there an easy proof?