Linked Questions

16 votes
6 answers
4k 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}{...
cm007's user avatar
  • 339
21 votes
3 answers
3k 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. ...
John Joy's user avatar
  • 7,840
14 votes
5 answers
2k 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 ...
Mike's user avatar
  • 13.3k
16 votes
4 answers
445 views

When is $8x^2-4$ a square number?

I asked an earlier question on when $32x+32$ is a square number (here) and I got a very clear answer. Now I am looking to solve for which $x$ the equation $8x^2-4$ results in a square number. When I ...
Héctor van den Boorn's user avatar
2 votes
5 answers
1k 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: ...
Stefan4024's user avatar
  • 35.9k
3 votes
2 answers
1k 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 ...
Redundant Aunt's user avatar
6 votes
2 answers
2k 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 $A$ ...
Yann's user avatar
  • 474
6 votes
2 answers
436 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 ...
James S. Cook's user avatar
0 votes
1 answer
253 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 ...
mr eyeglasses's user avatar
3 votes
2 answers
170 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=...
Sinister's user avatar
  • 585
0 votes
1 answer
187 views

Slightly more general method to check whether or not $ax^2+bx+c$ will ever generate a perfect square integer?

I have already looked at this post and this post but can't seem to apply completing the square to some cases because the numbers become non-integers. Given a quadratic equation of the form $$f(x)=ax^2+...
J W's user avatar
  • 13
1 vote
1 answer
167 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-...
Ewan Delanoy's user avatar
  • 61.6k
0 votes
2 answers
115 views

Seed solutions to a diophantine equation and Reversibility of the Conway's topograph method

I came across the answer to solving a quadratic diophantine equation on this site by @Willjagy: General method for determining if $Ax^2 + Bx + C$ is square I wish to know how the four seed solutions ...
RTn's user avatar
  • 299
1 vote
0 answers
38 views

Solution to an indeterminate equation

I have an equation of the form $$Dx^2-2s.t.x+t^2=c^2$$ where $s$, $t$ and $x$ are positive integers and $c$ can be any positive and odd integer. Is there a method to recursively find the values of $x$ ...
RTn's user avatar
  • 299