Linked Questions

75
votes
2answers
40k views

Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer

Let $a,b$ be positive integers. When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square. Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two ...
22
votes
4answers
4k views

No maximum(minimum) of rationals whose square is lesser(greater) than $2$.

Suppose $A$ is the set of all rational numbers $p$ such that $p^2 <2$ and $B$ is the set of all rational numbers $p$ such that $p^2 > 2$. We want to show that $A$ contains no largest element and ...
25
votes
4answers
2k views

Can Number Theory be visualized?

So I was thinking about a hard euclidean geometry problem, when it hit me just how much more difficult it would become without the aid of a diagram. This got me thinking: Wouldn't it be great if we ...
19
votes
4answers
2k views

Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer

The question is: Show that there are an infinite number of pairs $(m,n): m, n \in \mathbb{Z}^{+}$, such that: $$\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{Z}^{+}$$ I started off approaching this ...
13
votes
9answers
14k views

IMO 1988, problem 6

In 1988, IMO presented a problem, to prove that $k$ must be a square if $a^2+b^2=k(1+ab)$, for positive integers $a$, $b$ and $k$. I am wondering about the solutions, not obvious from the proof. ...
2
votes
2answers
21k views

Simple solution to Question 6 from the 1988 Math Olympiad [duplicate]

Recall Question 6 of the 1988 Math Olympiad Question 6 is as follows: Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Show that $$\frac{a^2+b^2}{ab+1}$$ is the square ...
32
votes
1answer
7k views

Math olympiad 1988 problem 6: canonical solution (2) without Vieta jumping

There is a recent question about this famous problem from 1988 on this forum, but I'm unable to respond to this because the subject is closed for me (insufficient reputation). Therefore this new post ...
8
votes
2answers
146 views

Show that infinitely many positive integer pairs $(m,n)$ exist s.t $\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{N}$ [duplicate]

Show that infinitely many positive integer pairs $(m,n)$ exist such that $\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{N}$. I couldn't solve it but I did make an observation which might or might not be ...
-2
votes
2answers
59 views

Two different polynomial expressions for certain primes

Here is an other random conjecture which I have no clue how to prove: $a,b\in\mathbb N^+\wedge a^2+b^2+ab\in\mathbb P\implies\exists$ $A,B\in\mathbb N^+:A^2+B^2-AB=a^2+b^2+ab$. Tested for $a,b<...
1
vote
2answers
85 views

Combinatorics problems that can be solved via infinite descent

I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a ...