Prove that 3a²-1 is never a perfect square

Answer :
We write 3a² -  1 , As :

= 3 ( a² -  1 ) +  2

= 3 k  +  2,  Here  k = a² − 1.
And we know  that the square of an integer must either be of the form 3 k or 3k + 1.

Hence, 3a²  -  1 = 3k + 2 cannot be a perfect square.

we can also check for different value of a , As :

At a  =  1 , we get

3a² -  1  =  3 ( 1 )² - 1  =  2 , That is not a perfect square .

Or
At a  = 2 , we get

3a² -  1  =  3 ( 2 )² - 1  =  11 , That is not a perfect square .
Or
At a  =  5 , we get

3a² -  1  =  3 ( 5 )² - 1  =  74 , That is not a perfect square .

So,

we can say 3a²  -  1 never be a perfect square.                                       ( Ans )