+0

# Can the product of a square number and 3 still be a square number? This will help me answer the "Is there a perfect cuboid?" question

0
83
3

Can the product of a square number and 3 still be a square number? This will help me answer the "Is there a perfect cuboid?" question

Guest Mar 6, 2017
Sort:

#1
0

I have tried it with the first 100 square numbers, this is going to be hard.

Guest Mar 6, 2017
#2
+4753
+1

$$\sqrt{3(0^2)} = 0$$

That probably doesnt help any ahaha...

hectictar  Mar 6, 2017
#3
+4753
+1

$$3a^2 = b^2$$

The question is, "Are there any integer values of a and b that make this equation true?"

Zero works but I want to find something besides zero.

I can rewrite it like this:

$$b = \sqrt{3a^2} \\ b = (\sqrt{3})a$$

b needs to be an integer, so what integer for a can make b an integer?

Well since √3 is irrational, the only possible thing that, when multiplied by √3 comes out with something rational will be √3.

√3 times any integer will just be that integer times √3.

You could multiply by √(27), but all that is doing is multiplying by (√3√3√3). There must be an odd number of √3's in a to cancel out the first √3. And if there's an odd number of √3's in a, then a is irrational too and not an integer.

I think it is impossible with anything besides 0.

hectictar  Mar 6, 2017

### 19 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details