Linked Questions

3
votes
2answers
133 views

Integer as sum of 6 squares.

Can every integer be written as sum of exactly 6 squares? I am also curious to know when an integer can be written as sum of exactly 8 squares. I know the problem is related to Waring-Hilbert ...
1
vote
1answer
191 views

Generalized Pell's equation

We know that Let $d$ be a positive square free integer and $r$ an integer satify $r^2+|r|\le d$. Suppose $x$ and $y$ are positive integers that satify $x^2-dy^2=r$. Then $\frac xy$ is a convergent to ...
1
vote
1answer
135 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 $y^2 \equiv 1\ \text{mod}\ 3$ ...
16
votes
3answers
245 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 $...
4
votes
3answers
254 views

Find all natural numbers $n$ such that $21n^2-20$ is a perfect square.

Find all natural numbers $n$ such that $21n^2-20$ is a perfect square. I have got the following solutions via programming: $n=1,2,3,9,14,43,67$ but how can I find these manually? How can I ...
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 ...
2
votes
2answers
153 views

How to prove that the roots of this equation are integers?

Let there be an equation $a^2 + 4ab + b^2 - 121 = 0$ where I want to prove that a,b are integers. Then I want to find whether there are integer values of $b$ for which $a$ is also an integer. Let us ...
0
votes
2answers
351 views

Solutions to Diophantine Equations

I am looking for integer solutions to the equation $$x^2 = 5y^2 + 14y + 1$$ I know that Pell's Equation is of the form $x^2 - ny^2=1$ and that there exist algorithms to solve this equation. I was ...
2
votes
2answers
619 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 ...
1
vote
1answer
98 views

Small integral representation as $x^2-2y^2$ in Pell's equation

Let $k$ be a "representable" positive integer, in the sense that $k=|x^2-2y^2|$ for some integers $x,y$. Does it necessarily follow that $k$ can also be represented with small parameters, i.e. $k=|u^2-...
0
votes
1answer
154 views

Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 adjacent ...
6
votes
2answers
126 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 $...
3
votes
2answers
145 views

Infinitely many systems of $23$ consecutive integers

Prove that there are infinitely many systems of $23$ consecutive integers whose sum of squares is a perfect square. My try: $$(n-11)^2+\cdots+(n+11)^2=23n^2+1012=23(n^2+44)=m^2$$ so $m=23k$ , $n^2=...
5
votes
2answers
275 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 ...
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 ...

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