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
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 is odd, then . Unfortunately, the number I have passes this test. Are there any others I can do?
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.
Can a number of the form (i.e. ) be a perfect square?
Any potential square root has to be of the form or . Squaring the first option gives . 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 .
Does this seem like a good way to go about this problem, or should I try something completely different?
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).