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

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## Comments

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TopNewest@Nihar Mahajan , @Sravanth Chebrolu , @Parth Lohomi , @Rajdeep Dhingra , @Archit Boobna, @Jon Haussmann Sir. Help!

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See RMO 2001 solution

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

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There is another method sir

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

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

Bound it: WLOG assumeLog in to reply

Infinite

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

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

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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.

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It's infinite.

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How? Proper Solution Please?

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

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It is infinte, let p=0

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no actually its finite

there are two pairs 3,11 and 11,3

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$p,q$ to be primes.

But this question does not specifyLog in to reply

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

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$p,q$ must belong to.

To be precise the question has not specified whatLog in to reply

@Vaibhav Prasad ???

Oh yes. If the original question was for primes it is finite, what do you sayLog in to reply

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@Harsh Shrivastava

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

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