A new PSE member with square reputation

Question

(For those viewing this, this is asked from the time when upvotes on question leads to +5 rep.)

A new Puzzling Stack Exchange member (reputation = 1) asked a good question.
He got a square number upvotes, and a square number downvotes.
(more Upvotes than Downvotes)
So, His reputation is a square number below 500.

Then he answered a question, and got good responses.
Again, he got a square number upvotes, and a square number downvotes.
(more Upvotes than Downvotes)
Again, his reputation now is a square number, but still not more than 500.

How many upvote and downvote he get after asking?
Then, How many upvote and downvote he get after answering?

Bonus Question : How if he answer first before asking?

Answer 1

In the beginning, when he asks he gets

64 upvotes and 16 downvotes ending up with 289 rep = 17$^2$ rep

and then after he answers the question, he gets

16 upvotes and 4 downvotes ending up with 441 rep = 21$^2$ rep

so we are done!! :)

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For the Bonus:

After answering he gets

49 upvotes and 25 downvotes ending up with 441 rep = 21$^2$ rep

and then after asking he gets

9 upvotes and 1 downvote ending up with 484 rep = 22$^2$ rep

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EDIT: HOW I FIGURED IT OUT

The riddle never said anything about having to use pencil and paper so I made an assumption and assumed that both numbers were less than 200. Then I wrote a short C++ program and tested all possibilities, ending up with both my answers. Total time taken: 4 min and 30 sec.

Answer 2

We first write the equation as

$5x^2 - 2y^2 = z^2-1$

We can solve this mathematically by using the fact that

all perfect squares are $0,1, 4\mod 8$.

This means we can rewrite the equation mod 8 as

$5 \cdot \text{(0,1, or 4)} - 2 \cdot \text{(0,1, or 4)} = \text{(0,3,7)}$

Checking all cases for the parentheses, we find that the only cases that work are

(0,0,0), (0,4,0) and (1,1,3)

The first and second solution mean

x is a multiple of 4, and y is a multiple of 2

and the third solution means

x and y are both odd.

Checking all perfect squares of these forms, we find the only possible solution is

64 upvotes and 16 downvotes, giving 289 reputation.

Similarly for the second part, the equation is

$10x^2 - 2y^2 = z^2-289$

Taking mod 8 we get

$10 \cdot \text{(0,1, or 4)} - 2 \cdot \text{(0,1, or 4)} = \text{(0,3,7)}$

giving the solutions

(0,0,0) (0,4,0) (4,4,0) (4,0,0) and (1,1,0)

The first four solutions mean that

x and y are both even

and the fifth means that

x and y are both odd.

Once again, checking all possibilities, we end up with

16 upvotes and 4 downvotes, bringing his reputation total to 441.