1. ## Perfect Square Test

Are there any tests you can perform to see whether an integer is a perfect square? The number I want to test is odd. The only test I know is: If $k$ is odd, then $k^2 \equiv 1 \mod 8$. Unfortunately, the number I have passes this test. Are there any others I can do?

2. Um, from a previously answered question, you can also check the number of divisors it has . If it's odd, then it's a perfect square, if it's even then it's not.

Not sure if that's the kind of test you're looking for, but it works

3. I don't know the number, though. I'll describe the problem below, but I DO NOT WANT THE ANSWER OR EVEN A SOLUTION; I just want some tests that I can perform and maybe a nudge in the right direction.

Problem:

Can a number of the form $200...009$ (i.e. $2\cdot 10^n + 9, n\in\mathbb{N}$) be a perfect square?

Any potential square root has to be of the form $10j+3$ or $10k+7$. Squaring the first option gives $100j^2+60j+9=20..09 \implies j(5j+3) = 10^m, m\in\mathbb{N}$. I think with a bit of work I could show that there are no integer solutions to that equation; however I'm not so sure I can do the same for $(10k+7)^2=20..09$.

Does this seem like a good way to go about this problem, or should I try something completely different?

4. Originally Posted by redsoxfan325
Are there any tests you can perform to see whether an integer is a perfect square? The number I want to test is odd. The only test I know is: If $k$ is odd, then $k^2 \equiv 1 \mod 8$. Unfortunately, the number I have passes this test. Are there any others I can do?
I do not understand the question.
What is wrong with extracting the Square Root?

If the square root is an integer, it is a perfect square.
That just seems to be the most efficient way to get the result.

5. I can't do that because I don't know the number. (I explain the problem in detail in my second post.) I know that it starts with 2, ends with 9, and has an arbitrary number of zeroes in between. The question: can a number like that ever be a perfect square (for any number of zeroes).

6. You're looking too far.

Hint : any integer is congruent mod $3$ to the sum of its decimal digits.

Now what are the squares, mod $3$?

7. Originally Posted by Bruno J.
You're looking too far.
Hint : any integer is congruent mod $3$ to the sum of its decimal digits.
Now what are the squares, mod $3$?
Brilliant!