Close to perfect square

Number Theory Level 4

$\large x^2+6x+y^2=4$

Find the number of ordered pairs of integers $(x,y)$ satisfying the equation above.

6 8 1 3 4 2 12

1 solution

Brian Charlesworth
Jun 8, 2015

This equation can be rewritten as

$x^{2} + 6y + 9 + y^{2} = 4 + 9 \Longrightarrow (x + 3)^{2} + y^{2} = 13.$

Now the only pair of perfect squares that add to $13$ are $4$ and $9,$ so we can have either

$(x + 3) = \pm 2, y = \pm 3 \Longrightarrow (x,y) = (-5,-3), (-5,3), (-1,-3), (-1,3),$ or
$(x + 3) = \pm 3, y = \pm 2 \Longrightarrow (x,y) = (-6,-2), (-6,2), (0,-2), (0,2).$

There are thus a total of $\boxed{8}$ ordered pair solutions.

(The explicit identification of the ordered pairs is unnecessary for the purposes of answering this question, but was done for the sake of thoroughness.)

Nice solution sir.Please try this one

Kalpok Guha - 6 years ago

FYI: once you know the square of the radius (in this case, 13), you can go to OEIS to get the answer.

Pi Han Goh - 6 years ago

That's interesting. If we designate $\theta(n)$ as the number of such ordered pairs for a given positive integer $n,$ I wonder what the upper bound for $\theta(n)$ is? Just as an example, we have $\theta(422500) = 56.$

Brian Charlesworth - 6 years ago

CHALLENGE MASTER NOTE: Interesting observation, can you make a note about it?

Pi Han Goh - 6 years ago

@Pi Han Goh – Sure, I'll do that later. I was also wondering the following: what is the largest integer that can be expressed as the sum of two distinct squares in only one way? There may be no such largest integer, but given that the probability that an integer $n$ can be expressed as a sum of two squares in more than one way increases as $n$ increases I thought this was a reasonable question to ask.

Brian Charlesworth - 6 years ago

@Brian Charlesworth – " two distinct squares in only one way?"

Do you mean " two distinct positive squares in only one way?

Pi Han Goh - 6 years ago

@Pi Han Goh – Yes, that was my intent. I suspect that there may be no such greatest integer, but a proof could be tricky. I'll add this question to my note on $\theta(n).$

Brian Charlesworth - 6 years ago

Close to perfect square

1 solution

0 pending reports