# Thread: elem. # theo - "prove never perfect square"

1. ## elem. # theo - "prove never perfect square"

Prove $3a^2 - 1$ is never a perfect square.
The book hints to use the fact: "The square of any integer is either of the form 3k or 3k+1," which we proved in a previous problem.

So how do you start?

Assuming $3a^2 - 1$ is a perfect square, it must be able to be written in the form of 3k or 3k+1.
Then what? How do you show it can't or NEVER can be?

2. Originally Posted by cassiopeia1289
Prove $3a^2 - 1$ is never a perfect square.
The book hints to use the fact: "The square of any integer is either of the form 3k or 3k+1," which we proved in a previous problem.

So how do you start?

Assuming $3a^2 - 1$ is a perfect square, it must be able to be written in the form of 3k or 3k+1.
Then what? How do you show it can't or NEVER can be?
i guess you could assume a is even and show it doesn't work, then assume a is odd and show it doesn't work either. that is, you can't simplify it to get it in that form

3. Ok, so by showing that there's no possible way to put it into the form would be like "let a=2s+1" ... ect ... which is not of the form 3k or 2k+1 and so on with "let a+2s" ... ? Is that enough?

But wait: how do we even know a is an integer at all? The problem never said so ... ?

4. Originally Posted by cassiopeia1289
Ok, so by showing that there's no possible way to put it into the form would be like "let a=2s+1" ... ect ... which is not of the form 3k or 2k+1 and so on with "let a+2s" ... ? Is that enough?

But wait: how do we even know a is an integer at all? The problem never said so ... ?
yes, if you can show that, it is enough, because there are no other options. you have exhausted all possibilities. a is an integer, it has to be even or odd, and that covers all integers. so you would prove that it can never work no matter what

5. Originally Posted by cassiopeia1289
Assuming $3a^2 - 1$ is a perfect square, it must be able to be written in the form of 3k or 3k+1.
Because $3a^2 - 1 = 3a^2 - 3 + 2 = 3(a^2 - 1)+2$.
Therefore it has the form $3k+2$ not $3k,3k+1$.

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### 3a^2-1 is never a perfect square

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