How to Solve this?

How Many Ordered Pairs For $p,q$ exist if

${p}^{2}+7pq+{q}^{2}$ is the Square of an Integer?

EDIT:- p,q are Reals Note by Mehul Arora
6 years, 2 months ago

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- 6 years, 2 months ago

See RMO 2001 solution

- 6 years, 2 months ago

Set $p=q$ or set $p=8q$ shows that there's infinite number of solutions.

- 6 years, 2 months ago

There is another method sir

- 6 years, 2 months ago

Yes, set $p=0$ or $q= 0$ or $q = 8p$.

- 6 years, 2 months ago

Exactly sir.

- 6 years, 2 months ago

But how you got p = 8q one thing ?

- 6 years, 2 months ago

Bound it: WLOG assume $p,q>0$. $(p+q)^2 = p^2 + 2pq + q^2 < p^2 + 7pq + q^2 < p^2 + 8pq + 16q^2 = (p+4q)^2$, then $p^2 + 7pq + q^2 = (p+2q)^2 \text{ or } (p+3q)^2$.

- 6 years, 2 months ago

Infinite

- 6 years, 2 months ago

I think I have solved this before. Anyway , I have a solution but it works only if $p,q$ are prime positive integers. $\ddot\frown$

- 6 years, 2 months ago

Can you tell me your method. $\ddot \smile$

- 6 years, 2 months ago

Well , if you see the official solution of this question in RMO , they have done by completing $(p+q)^2$ whereas i did it by completing $(p-q)^2$. The rest of the method to get the answer is same but only my method has more cases since i have $9pq$ whereas the official solution has $5pq$. The advantage of official solution is that $5$ is a prime.

- 6 years, 2 months ago

It's infinite.

- 6 years, 2 months ago

- 6 years, 2 months ago

I did initially thought that there are finite solutions, but after seeing @Pi Han Goh sir's solution I was convinced.

- 6 years, 2 months ago

It is infinte, let p=0

- 6 years, 2 months ago

no actually its finite

there are two pairs 3,11 and 11,3

- 6 years, 2 months ago

But this question does not specify $p,q$ to be primes.

- 6 years, 2 months ago

HOW? I SAW THAT IN THE RMO SOLUTION AS WELL!

- 6 years, 2 months ago

there could be infinite.

actually the question was for primes. u r asking for positive integers.

- 6 years, 2 months ago

To be precise the question has not specified what $p,q$ must belong to.

- 6 years, 2 months ago

Oh yes. If the original question was for primes it is finite, what do you say @Vaibhav Prasad ???

- 6 years, 2 months ago

yes u r correct

- 6 years, 2 months ago

BTW, has it really appeared in RMO?

- 6 years, 2 months ago

RMO 2001

- 6 years, 2 months ago

Yeah....

- 6 years, 2 months ago

@Harsh Shrivastava

- 6 years, 2 months ago

Yes??

- 6 years, 2 months ago

Ar you sure you want "p,q are Reals" instead of "p,q are integers"?

Staff - 6 years, 2 months ago