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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
2 years, 10 months ago

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

Staff - 2 years, 10 months ago

It's infinite.

- 2 years, 10 months ago

- 2 years, 10 months ago

It is infinte, let p=0

- 2 years, 10 months ago

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

- 2 years, 10 months ago

no actually its finite

there are two pairs 3,11 and 11,3

- 2 years, 10 months ago

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

- 2 years, 10 months ago

there could be infinite.

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

- 2 years, 10 months ago

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

- 2 years, 10 months ago

yes u r correct

- 2 years, 10 months ago

BTW, has it really appeared in RMO?

- 2 years, 10 months ago

Yeah....

- 2 years, 10 months ago

@Harsh Shrivastava

- 2 years, 10 months ago

Yes??

- 2 years, 10 months ago

RMO 2001

- 2 years, 10 months ago

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

- 2 years, 10 months ago

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

- 2 years, 10 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. $$\ddot\frown$$

- 2 years, 10 months ago

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

- 2 years, 10 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.

- 2 years, 10 months ago

Infinite

- 2 years, 10 months ago

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

- 2 years, 10 months ago

There is another method sir

- 2 years, 10 months ago

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

- 2 years, 10 months ago

But how you got p = 8q one thing ?

- 2 years, 10 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 $$p^2 + 7pq + q^2 = (p+2q)^2 \text{ or } (p+3q)^2$$.

- 2 years, 10 months ago

Exactly sir.

- 2 years, 10 months ago

See RMO 2001 solution

- 2 years, 10 months ago